ch 7a measurement concepts

7/30/2019 Ch 7a Measurement Concepts

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Measurement Concepts and Measuring Atributes with Instruments Ch. 7, page 1

Measurement Concepts and Measuring Attributes with

Instruments

Students Understanding of Measurement

A report on primary school students understanding of science and technology in

Australia (Pattie, 1995) revealed that the effect of teaching was evident for the sample

of 1161 children from 34 government and non-government schools. The study indicated

that students performed at the higher levels when measuring temperature, length, and

mass, reading a classification chart, and using space relationshipsmathematical skills

used in science and technology. However, such skills need to be further related to the

scientific context. There was no overall difference in performance between males and

females, or between children in urban, suburban and rural schools but there was a high

correlation between socio-economic status and childrens understanding of science

concepts and their performance of science skills.

Measurement and Conservation

Piaget originally thought that since pre-operational thinkers cannot conserve, then

certain learning experiences for measurement should be delayed. It was previously

considered that students could not explore the attribute, measure directly or indirectly

until able to conserve but experience in classrooms clearly indicates that the experiences

assist students to conserve.

Childrens Development

Piagets early research suggested that conservation of length, area, volume wereissues in learning. He felt volume was much later than length and area.

Later research suggested that informal play with capacity (filling containers with

sand or water) showed that conservation of area is more difficult than expected and that

volume (cm3 ) is more difficult than litre capacity.

Mass is also quite difficult to sense without hefting in your hands and is visually

confused with volume or capacity. A large, light object has a low density, and it has a

low mass. However, a small, heavy object has a high density, and it has a large mass.

Experiences with different types of objects is important.

Later research suggests that the issues of conservation were clouded by perceptual

dominance and experience and that measuring as a concept needs to link to everyday,situated learning e.g. ruler.

Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics Kay Owens


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Learning Tasks for the ReaderSelf-check on Conservation

experiencing



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Generic Development of Measurement Concepts

No matter what attribute of measurement, certain critical aspects of measurement need

to be developed. These are:

Recognition of the attribute

The different representations of the attribute,

e.g. for length, we have length, height, width, distance, curved line, perimeter

e.g. for area, we have flat, horizontal surface, curved surface, combined surfaces

The idea of measuring

comparing size directly

comparing size indirectly

needing to be more versatile, precise, non-visual

The idea of a unit

The idea of a composite unit made of joined units

The idea of different base units

Count Me In Measurementprovides six levels of development. The project provides

lessons for each level for each attribute: length, area, mass, capacity and volume.

Level 1: Identification of the attribute includes directly comparing and ordering

quantities

Level 2: Informal measurement includes finding the number of units to cover, pack or

fill a given quantity without overlapping or leaving gaps; knowing that the number of

units used gives a measurement of quantity; using these measurements to comparequantities and realising that a quantity is unchanged if it is rearranged (the principle of

conservation)

Level 3: Unit structure includes replicating a single unit to cover, pack or fill a given

quantity, either by drawing or visualising the unit structure; and realising that the larger

the unit, the fewer units will be needed

Level 4: Recognise, measure and record in conventional units

Level 5: Use relationships between units and from the geometry to measure and

calculate in smaller and larger units

Level 6: Knowing and representing large units, consolidating and converting units, usescale

Learning Tasks for the Reader

Measurement Activities 1

experiencing Find an interesting object in the room, measure different aspects

of it and in different ways. What were critical aspects of your

measurement?

How many people can stand in the room? Compare the different



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ways people do this and also think about short-cuts, especially

consider the natural units in the room.

Make circular, triangular, square, other prisms with the lateral

faces made by folding an A4 sheet of paper.

Describe which prism and why has the largest volume from A4

paper.

Show the length of string which is equal to the average height of

your group of three

Draw different shapes with an area of 12 square units. What did

you consider?

Make stacks of 24 cubes. Try making some stacks which are not

rectangular prisms. Make them with different surface areas.

Select a surface area and make stacks of different numbers of

cubes.

connecting ideas

What did you have to know about to measure an object? Was it

the only thing that you could have measured about the object?

What does that tell you about attributes of objects and

measurement?

The area of the room could be measured using rectangular units.

What does that tell you about units? What does it tell you about

the development of the formula for the area of a rectangle? What

did you learn about composite units if you tried short-cuts?

What did you learn about the number of units needed when you

measure with smaller units?

Look at the three attached diagrams of squares on grids. Discuss

how these require an understanding of a grid and the idea of

composite units.

Was the comparing of the volume of different prisms, a taskabout area or volume? Square centimetre paper could be used

for greater accuracy. How accurate would the measurement of

the base of the prisms be?

The average height activity helps in developing the concept of

addition of lengths and the meaning of average. Explain.

Can a space measure a square unit but not be a square? What do

these activities tell you about shapes, the distinction between

shape and area, and the effect of perception on estimating area?

Do all the shapes for 12 square units have to be polygons? Whyand why not? How might the variance in shape for 12 square



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units be used in advertising or packaging?

What do you learn about formulae for area and volume from the

tasks?

summarise

and record

What previous experiences helped you with your

approaches?

Did you change your procedure/thinking during the

problems? Why?

What do these experiences tell you about the value of

problem solving?

What concepts did you use in finding the solution? How did

you refine your understanding of these concepts?

What do you think an area unit is?

What might be meant by composite units of area?

LengthBoulton-Lewis, Wilss, & Mutch (1994) asked students to measure the length of two

lines made from joined matchsticks; each configuration had a recognisable pattern.Younger students were likely to choose the familiar ruler rather than the unfamiliar

measurement units (sticks) in order to attempt to compare the lengths of the two lines.

Boulton-Lewis et al. suggested that the idea of introducing measurement with arbitrary,

informal units may not be appropriate for students if they are to grasp the concept of

measuring length because they do not have a mental model based on familiarity and

past experience of the arbitrary units. By contrast, many syllabus documents have

suggested that there be experiences with informal units before formal units. Willis

(2005) suggests that there is much more about using units than what type. To begin with

a unitis an abstract idea. The stick is only representative of that idea. Similarly, when

talking about gaps and overlaps in tiling for area, the tile (even if it is 1 square

centimetre) is a thing not the idea of an area unit that could take any shape. Earlyresearch also showed that young children could recognise that they would need fewer of

a larger unit than a smaller unit to measure a fixed line.

AreaThere have been several studies on various issues related to this concept. Doig,

Cheeseman, and Lindsey (1995) investigated the effects of different materialpaper

squares, Dienes blocks and wooden tileson children's success in measuring the area

of a rectangle. They found that children were least successful when they used the paper

squares and most successful when they used the wooden tiles. Children who used

Dienes blocks were most likely to confuse measurement of area with that of perimeter

or length. However, the use of paper squares revealed inadequate understandings of area

because students were more likely to overlap or leave gaps between the paper squares.

Consequently, practice in tiling with rigid materials may not help students



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understandings of area (Outhred, 1993). If mathematical concepts are to emerge, it

seems important that concrete experiences of covering areas should engage students

visual imagery and analysis (Owens, 1994b) and should also involve student-student

and student-teacher interaction about the ideas needing development (Hart & Sinkinson,

1988; Owens, 1994a).

Clements and Ellerton (1995) interviewed a large sample of students on several testitems used in basic skills tests in Australia. They showed that there were notable

proportions of mismatches between correct/incorrect answers and non-

understanding/understanding as probed during interviews. One of the items was to find

the area of a trapezium consisting of a square and a triangle (half the size of the square)

with only some lengths given. The study showed a large number of students did not

understand the concept of area, how lengths relate to area, why different shapes have

different formulae for calculating, or the value of visual/spatial knowledge. In another

study Clements (1995) illustrated a lack of conceptual understanding for a student able

to calculate the area of a triangle.

Young children often hear the word area referring to place, and may think of area assomewhere to gofor example, the assembly area or the reading areawithout

considering it as a region. They do not seem to realise that such regions are two-

dimensional (2D) spaces enclosed by boundaries and that they can be covered with units

(e.g., sheets of newspaper).

Students may have covered small regions such as desks, books, and chairs with

informal units and perhaps compared the two by counting the number of units needed to

cover them. Such activities, intended to be introductory to the concept of area

measurement, may in fact confuse students. The use of irregular shapes and informal

units (e.g., potato prints) may result in the activity being dominated by counting, while

ideas crucial to the concept of area measurement (e.g., overlaps, gaps, and congruent

units) are ignored (Outhred, 1993; Willis, 2005). Willis emphasised that students whohad counted to decide on the measure of an objects length or mass were then unable to

use this information to answer a question about whether the object was heavier than

another or longer than another. A greater appreciation of the concept of covering would

seem to be necessary if older students are to calculate areas meaningfully (see

Mitchelmore, 1983, and Clements, 1995, for examples of typical student difficulties

with area calculations).

In observations of pre-school children covering squares, rectangles, and triangles

with smaller cut-out rectangles, squares, and triangles, Mansfield and Scott (1990) have

shown that students vary in their ability to choose appropriate unit shapes, in their

persistence, and in their turning and flipping tactics. The most difficult shape for the

children to cover was an equilateral triangle with a point facing down. Familiarity with

the shape to be covered seemed to increase success on the task.

In a study by Wheatley and Cobb (1990), students were asked to cover a large

square by selecting shapes from a collection comprising a square, several triangles, and

a parallelogram. Some students chose only the parallelogram and tried to cover the

square with it, an approach which suggested to Wheatley and Cobb that the students

were matching lengths. Alternatively, students may have chosen the shape that appeared

to be largest. Other errors included leaving gaps, especially on the sides, and

overlapping pieces or the sides of the square.

Drawing may be one way of linking experiences with concrete materials to students

mental models of tessellations. Several researchers (e.g., Mitchelmore, 1983; Outhred,1993; Outhred & Mitchelmore, 1992) have suggested that drawing is an important tool



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in developing students knowledge of rectangular arrays and in making links to

multiplication. Outhred (1993) found that many students had difficulties visualising or

drawing tilings of square units to cover rectangles when the squares were only shown

on adjacent sides of the rectangle or indicated by side marks, particularly for rectangles

with large dimensions. Some students drawings suggested that they did not understand

what features of arrays were important in constructing tessellations of squares. Owens(1992, 1993) found that students in Years 2 and 4 had difficulties imagining tilings of

squares, rectangles, and triangles to cover larger shapes. For example, in the activities

illustrated in Figures 1a and 1b, they had difficulty in predicting the number of smaller

triangles that would be needed to cover the larger ones. Very few students commented

on the amount of space covered when asked what was the same about different

arrangements of five squares (pentominoes); nearly all focused on the number of tiles

(Owens, 1993).

(a) Tangram triangles (b) Pattern-block triangles

Figure1. Shapes made during spatial activities (Owens, 1993).

The studies mentioned above, especially those by Outhred (1993) and Owens

(1993), emphasise the importance of spatial thinking and visualising when students

cover and compare shapes. To learn about tiling, students need to identify suitable units,

to transform shapes to other orientations, to recognise and partition shapes, and to

identify key features of shapes (e.g., matching parts such as right angles or equal

lengths).

Owens (1993, Owens & Outhred, 1998) examined the drawings of students in

Years 2 and 4 who were asked if specific units (squares, rectangles, right-angled

triangles, and equilateral triangles) could be used to tile figures and how many units

would be needed. Three factors that seemed to influence children's responses were

summarised by Owens and Outhred (1998):

1. Size of tiles. While children seemed to know that there was a pattern for filling

the space, some seemed to retain the shape but not the size of the tiling unit. Children

who drew tessellated tiles without regard for size usually felt that the space was beingadequately filled.

2. Recognition of tile features. Students frequently used the sides and corners to

begin filling in spaces with tiles. The type of corner seemed important to some students

in deciding if a particular tile would be likely to fit. Students had difficulty recognising

a trapezium as a composite of tessellated right-angled triangles, despite prior experience

with concrete materials.

3. Judgments about drawings. Some students decided that overlaps or size or gaps

did not matter if the result was "close" enough, that is, they based their judgments on

their spatial sense and ignored slight discrepancies in their drawings. Children who

thought the tiles should fit often filled gaps with additional tiles, disregarding shape or



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overlap. Others were influenced by inaccuracies in their drawings and said that the

shapes could not be made by fitting tiles together.

While Owens was carrying out her study, Outhred (1993) was independently

exploring children's difficulties in representing arrays and how such difficulties are

related to performance on area measurement tasks. Her research suggests that

knowledge of array structure provides a link between measurement and multiplicationconcepts in the context of rectangular area measurement. She found that knowledge of

array structure was essential for children to relate the lengths of adjacent sides of a

rectangle to the number of squares that would cover it. These results indicate why

activities with concrete materials may not be sufficient to help children understand the

formula for the area of a rectangle. When measuring the area of a rectangle using

concrete materials children do not require awareness of row and column structure

because the structure is determined by the materials, rather than by the child's thinking.

The effects of specific types of instruction on children's use of lines to represent

rows and columns was investigated with children in Years 1 and 2 (Outhred, 1993). The

findings suggested that teaching children that squares in an array were all the same sizewas not the most effective method to help children to perceive array structure. Teaching

children that the squares are aligned or that there are the same number of squares in

each row (column) seemed to be more effective methods for moving children from

drawing squares individually to representing rows (columns) of squares using lines.

McPhail (1997) has continued to explore how young students develop knowledge

about tessellations and area. With a series of four lessons she has shown that young

students in Years 1 and 2 can learn about area. Her lessons allowed students to first

make their own area enclosed by a length of braid. The children also painted or rubbed

large triangles and squares which were later used to make large visual displays of

tessellations. In addition, the students had many small squares and triangles. They were

asked to make large squares and triangles as well as covering given ones. The cardboardtiles had the edges in black so that arrays were easily seen. This seemed to facilitate

children drawing tilings using arrays rather than individual tiles. Interestingly, the

children applied many number facts in telling the teacher how many tiles they had used.

Some used the ideas of repeated addition of rows of tiles, others counted a number and

then added on the subitised remaining number of tiles. Students could explain the

patterns of the triangular covering.

Willis (2005) provides another challenge in saying that a tile is a measuring

instrument. Rigid tiles for area may not cover a thinner object without discussing

cutting up the tile. Unlike the fluid units used in capacity that flow around and fill the

container, this does not automatically happen with area. They concluded that

understandings should include:

(a) the instrument we choose to represent our unit should relate well to the attribute

to be measured and be easy to repeat to match the thing to be measured;

(b) to measure consistently we need to use our instrument in a way that ensures a

good match of the unit with the object to be measured;

(c) units are quantities and so we can use different representations of the same unit

so long as we do not change the quantity (Willis, 2005).

Volume

Toh (1993) compared the effectiveness of computer simulated experiments with

that of parallel instruction involving hands-on laboratory experiments for teaching

volume-by-displacement concepts. The purpose of the simulation was to have students



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test their misconceptions rather than simply being told about erroneous misconceptions.

The study consisted of 389 students from 6 Malaysian schools. The results indicated

that the computer-assisted group was significantly better in terms of learning gains in

the cognitive categories of knowledge and application. The computer simulated

different conditions such as same shape, different masses; or same mass, different

shapes. While it is not easy to do this with concrete materials, nevertheless theexperience assists in moving students on from limited conception about the volume by

displacement.

In summary, teachers

should be aware of the confusion between area and perimeter because students

do not develop the concept of area, they do not develop area formulae, or they

are just told to use the formula (with the meaningless idea of lengths becoming

area rather than the formula involving numbers only without the unit attachment

and that it is a formula associated with a specific shape, e.g. rectangle);

can establish the area concept through painting an area, tiling, discussing no

gaps, and the nature of shapes; should not use just solid tiles that structure exercise and prevent abstraction, or

lead to counting exercise rather than an area experience;

should know the value of tiles that are not square;

should know the value of recognising patterns and drawing grids (and discuss

drawing difficulties);

should recognise the importance of visualising and estimating;

should build on childrens intuitive dissecting of areas to assist in calculating.

Units

Students need to develop concepts like area that are measured. Spatial experiences

and knowledge about shapes will help when comparing informally or directly or when

selecting a unit for measuring. Later knowledge of the properties of shapes will assist

students to calculate areas.

We also use standard units so that there is no confusion. Students need to be

familiar with these, to know about how big they are, and to be able to estimate in these

units. In particular, students need experiences in (a) selecting objects that represent the

unit, (c) estimating, and (d) measuring in order to develop a sense of these units.

Composite Units

An important concept in measurement is that of a composite or iterable unit. For

example, when the young students put tiles in rows and count by rows, for example, 3,

6, 9, they are using the row as a composite unit made up of 3 units. This idea is also a

basic concept in both multiplication and our place value system with a ten being a

composite unit of 10 ones.

While students first develop a sense of units which are within their grasp, they

expand these into larger and smaller units using the notion of composite unit. Students

eventually need to be able to change from one unit into another. A good understanding

of the composite unit will assist this procedure.



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Metric Units and the Place Value System

Measurement can assist students to develop their understanding of the place value

system. For example, students can read off the length of an object from a ruler in (a)

metres, (b) metres and centimetres, and (c) centimetres. The ruler can establish the idea

that a metre is a composite of 100 centimetres but also that a hundred is a composite of

100 ones. More importantly, the idea of one being a composite of 100 hundredths is

also established. So, for example, a string might be 1.23 m or it can be written as 123

centimetres or 1 m 23 centimetres. Activities allowing for these various descriptions

will assist students to make the links. They will take time.

Recognising Structure

Students need to recognise structure in order to develop their measurement

concepts. Mulligan expresses this in the following diagram. See your earlier activities

on area of a room, making a ruler and area sheets for units.

Learning Tasks for the ReaderSelf-check on Measurement Sense

experiencing1. At what temperature does a person suffer from hypothermia?

a. 25o

b. 30o

c. 35o

d. 40o

e. 45o

2. The height of a child in Year 2 is about

a. 1 cm

b. 50 cm

c. 100 cm



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d. 150 cm

e. 1 000 cm

3. A small fish tank would hold about

a. 1 L

b. 20 L

c. 100 Ld. 1 mL

4. The area of floor in front of the desks is about

a. 1 000 m2

b. 100 m2

c. 10 m2

d. 0.5 km2

5. A house block has a house with gardens covering the same area

as the house. The area of the block is about

a. 0.5 hectares

b. 1 hectarec. 1.5 hectares

d. 2 acres

6. An A4 sheet is about

a. 10 cm2

b. 100 cm2

c. 1 000 cm2

d. 10 000 cm2

7. The space taken up by an engine of a small car is about

a. 1.4 L

b. 1.4 m3

c. 1.4 m

d. 1.4 kg

8. A litre of water has the same mass as:

a. a commonly available bag of rice

b. a house brick

c. a can of condensed soup

d. a dozen eggs

e. 2 apples

Think about this problem. If you hold a mans handkerchief

diagonally, how many are needed for the length of a horse?

Before wheelie bins, how high was the garbage bin?

connecting ideas

What role does a sense of size play in measurement?

How did (a) visualising; (b) experience with the unit; and (c)

prior experience impact your decision-making?

What influences the necessary degree of accuracy?



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What are some ways that you can encourage students to develop

a good sense of volume and area?

Why should students have many activities to encourage them to

construct both a sense of area and formulae for the area of a

rectangle and a triangle, and the volume of a prism.

Why are square units so useful?

summarise

and record

Look in the Syllabus at the experiences that students need for

developing their own area formula for a rectangle, a rectangular

prism and a triangle

Include in your summary for area comments on:

(a) the importance of number of rows and number of square

units per row

(b) how to deal with sides that are not whole numbers (e.g.

folding rectangular areas)

Include in your summary for volume comments on:

(c) Number of layers and number of cubic units in each layer

(d) The link between 1 cm3 and 1 mL.

Outcomes for NSW Mathematics K-6 Syllabus

Table 1 gives the NSW outcome statements for measurement.

Table 1

Outcome Statements for Measurement

Early Stage 1 Stage 1 Stage 2 Stage 3

Length MES1.1 Describes

length and distanceusing everyday

language andcompares lengths

using directcomparison

MS1.1 Estimates,

measures, comparesand records lengths and

distances usinginformal units, metres

and centimetres

MS2.1 Estimates,

measures, comparesand records lengths,

distances andperimeters in metres,

centimeters andmillimeters

MS3.1 Selects and uses

the appropriate unit anddevice to measure

lengths, distances andperimeters

Area MES1.2 Describesarea using everyday

language andcompares areas

using direct

comparison

MS1.2 Estimates,measures, compares

and records areas usinginformal units


and records the areas ofsurfaces in square

centimeters and square

metres

MS3.2 Selects and usesthe appropriate unit to

calculate area,including the area of

squares, rectangles and

triangles

Volumeand

MES1.3 Comparesthe capacities of



MS3.3 Selects and usesthe appropriate unit to



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Capacity containers and the

volumes of objects

or substances usingdirect comparison

and records volumes

and capacities using

informal units

and records volumes

and capacities using

litres, milliltres andcubic centimeters

estimate and measure

volume and capacity,

including the volumeof rectangular prisms

Mass MES1.4 Compares

the masses of two

objects anddescribes mass

using everydaylanguage

MS1.4 Estimates,

measures, compares

and records the massesof two or more objects

using informal units

MS2.4 Estimates,

measures, compares

and recordsmassesusing kilograms and

grams

MS3.5 Selects and uses

the appropriate unit and

measuring device tofind the mass of objects

Time

Passage of

time, itsmeasurem

ent and

representations

MES1.5 Sequences

events and uses

everyday languageto describe the

duration ofactivities

MS1.5 Compares the

duration of events

using informal methodsand reads clocks on the

half-hour

MS2.5 Reads and

records time in one-

minute intervals andmakes comparisons

between time units

MS3.5 Uses twenty-

four hour time and am

and pm notation inreal-life situations and

constructs timelines

Measuring Instruments

Students need to understand measuring instruments. For example, the scale on a jug

looks like a ruler for measuring length but it is indicating volume. How can we get

students to understand that? Most rulers have gradations that are equally spaced, i.e. 1

and 2 are spaced at the same distance as 2 and 3. This is not always the case depending

on the purpose of the instrument. Students also have to understand how to read the

gradations that are not marked and they need to know that the number is not the point

but the measure from the start. For example, it is the amount of water in a cup; at zero

the cup is empty.

What Instruments can we Make for MeasuringIf students are going to appreciate how a ruler works and that the numbers on the

ruler are representing the length from the start of the ruler, then they need to make a

ruler. It can be done by lining up some base 10 long blocks and marking off and

numbering a strip of paper, cloth or plastic. Alternatively they can make lengths of 1

cm, 2 cm etc, put each length on their long strip of paper with the starts together and

marking the length and writing down the length on their long strip.

Figure 1. Lengths of paper being used to make a ruler.

A volume measure can be made by using a jar and lids. As the student puts in

another lidful, they mark where the water comes to and the number of lids now in the

jar. This helps students to see the linear marks on the measuring jar as representing

volume.

Students can bring to school all sorts of measuring instruments that can be found in

the home and garage. All sorts of gauges are used. These can be discussed even if the

students do not fully understand what is being measured. They can see the needle

moving through the numbers on bathroom scales; they can look at the different widthsof the sparkplug gauges; they can watch an amp metre needle swing.


0 5


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Students can make various time clocks.

Learning Tasks for Readers

Making Measuring Instruments

experiencing

Make a pair of calipers to pinch the fat on your back to measure

fat.

Make a tapered diameter measure to see how big a ring you

make when you touch your thumb and forefinger.

Make a tiny trundle wheel to measure around your leg or waist

in cm (the wheel can have a circumference of 10 cm.)

Makea balance - spring balance or an equal arm balance.

connecting ideas

What is a measuring instrument?

Outline some important early and later experiences that students

should have for establishing the concept of mass,

summarise

and record

What important concepts in measurement should students learn?

What are important concerns in teaching about measurement?

Using Measuring Across the Curriculum

Clearly measuring is an important skill in Science and Technology. Students

should investigate in science and use measuring as a tool. For example, take different

brands of nappies and investigate which is the best.

For older students, two great tasks are from MCTP Activity Banks (Lovitt &Clarke, 1989) called Danger Distance and Map of Australia. Both encourage

visualisation with measurement.

Finding old measuring instruments around the farm, old mine site or dump,

designing and making measuring instruments can also be fun.

Measuring and reading measurements and interpreting data are all important ideas

in investigating nature and studying, for example, animals.

Human Society and Its Environmentprovides plenty of opportunities for using

measuring and interpreting information. Map work and built environments are just two

areas.

ForPersonal Development, Health, and Physical Education, there are manyideas. What does it mean to have a pulse rate of 60 beats per minute? What does it



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mean to have diversity in height at a particular age group? What does it mean to have

10g of fat per 100g, compared to 1g of fat per 100g?


Mathematics and the Human Body

experiencingWhat is our lung capacity?

What is meant by lung capacity?

How can we find out the volume of air in our lungs?

There are machines which can be discussed.

One approximate way is to blow up a balloon with one

breath.

Discuss how you can get the volume of the balloon. One way

is by putting the balloon in a full bucket of water and

measuring the displacement of water.

Remember that fit people and non-smokers improve their

lung capacity. Lung capacity grows from childhood to

adolescence.

How much skin do we have?

Estimate how manysq cm for the sole of your foot.

What is the skin mathematically and what is meant by the size

of our skin?

A discussion on this questions should be about covering of the

surface and surface area. Areas can be in different forms

including surface areas and this can be modeled by wrapping

with newspaper.

Discuss how to find the area of different parts of the body.

Arms can be represented by curved cylinders which can be

flatten out to rectangles (or near rectangles). Discuss ways of

measuring the rectangles with informal square units.

If you add up the size of all the parts of the body surface in sq

cm and try to convert to sq m, ask yourself whether the answer

seems sensible. How do you convertsq cm tosq m?

A rule-of-thumb for estimating the total amount of skin is to

multiply the size of the sole of the foot by 100. This is then a

quick way of deciding on the percentage of skin burnt on a burn

victim. Try out this rule-of-thumb and decide what percentage

of skin an arm would be.

Compare the sizes and ratios for a small child and for an adult.

connecting ideas

Discuss the following suggestions for activities. Think about:

the equipment that might be needed,

the different approaches that might be taken,

questions that students may ask and how to answer theirquestions, and



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how to get them to investigate.

How much carbohydrate food is in a packet of crisps? How

much in a potato? Which is better food?

(Health note: We need quite a bit of carbohydrates with fibre but

not much fat each day.)

Students undertaking this investigation may:

- investigate the nutrition information

- ask about why grams are used, learn that g is the symbol for

grams

- weigh potatoes, decide what an average one might be

- compare the amount of oil in a packet of crisps by weighing

that in cooking spoons

Do we drink enough water?

(Health note: Children should drink about 8 cups per day.)

Compare drinking bottles, glasses, and other drink containers.

Discuss how we can compare - cups, L, mL (depending on

age).

(Teachers need to think about how they can measure bubbler

drinks.)

Compare the different shapes of containers with the same

amount of water.

Students can extend the activity and measure more accurately.

Discuss how the perceptions of size are used in marketing and

how different shapes help storage and handling. And here we

can link in the eyes and the brain interpreting what we see and

the effects of stored information.

How big is your heart?

The heart is said to be as big as the persons fist.

Think about and try out how to get its volume (by displacement of

water).

How flexible is our heart and how does it respond to help us?

Find your pulse and count how often it beats. Do this when you

enter the class after running around at recess. It is usually a little

stronger and it will be easier to find, especially at the neck. Then

discuss how to count it.

Discuss the idea of rate. It is not an easy idea but really

important for some discussion in primary school as it is a major

idea in later studies. One way of doing this is to show that inquarter of a minute is quarter the number in a full minute. The



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ratio is the same.

Take the pulse rate after sitting for awhile and then after some

vigourous activity the same for the whole class.

Compare different pulse rates and discuss how adaptable the

heart is to meet their needs.

If some studnets are regular swimmers, exercise regularly, or

run around more than others, you might be able to compare their

slower rates after exercise (or how much quicker their rate

returns to normal). Discuss the effect of fitness on heart rate.

summarise

and record

Why do students learn more about mathematics through:

investigations and

real-life contexts?

How do you make sure that you are covering mathematics and

other Key Learning Area outcomes when giving students

investigations like those listed above?

Can these activities be modified for different age groups? If so,

give some examples.

Planning Events, Times and Calendars

The ancient Babylonians were keen astronomers, astrologers, and travellers. They

made links between the number of days in a year and the time it took for a cycle of

seasons to pass and the earth to rotate around the sun.

We have 365 days as closer to the time taken to complete the cycle but the number360 is more useful because it can be divided up into many different ways.

Calendars vary from place to place and culture to culture. Yearly and daily periodsof time are described differently in different cultures. A diagram that shows how one

Indigenous Australian tribe describes the time of the year shows the close links between

times and knowing when to fire the grass so that bushfires do not start and when to get

certain food. The overlap of events is easiest represented by segmented concentric

circles. One circle represents the wind seasons, another the plant seasons and so on. Any

period can be determined by the coincidence of these events (Harris, 1989).

Planning Events and Feasts. Many Pacific Islanders organise large feasts. There

is much mathematics involved in deciding on the quantity of food to prepared in theMarae kitchen, how many bundles (about one for every 50 people) are needed and how

to go about gathering the food together. For people who are growing the food, months

of preparation is involved in planning gardens so that the food is ready for the feast.


Planning Events, Times and Calendars

experiencingDo you know why we have 360o in a circle?

Whatnumbers go exactly into 360?

What events or gatherings do you plan? What mathematics do



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you use when you plan for these events?

connecting ideas

Look closely at the lessons on time in the Syllabus. Compare this

with the set of outcomes in Table 1.

Which has greater emphasis in Kindergarten: (a) recognising theattribute of time by comparing the time taken for events, or (b)

reading the clock.

What are some difficulties that young students will have reading

analogue clocks?

summariseand record

Summarise the new ideas that you have learnt about

measurement, especially time, as a result of considering the

differences in cultures.

Summarise the ideas of

attribute composite units

culturally determined measures

metric system

links with decimal place value

C i S P f i l K l d d S i l A i i i f T hi M h i K O

ch 7a measurement concepts

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