1 Building-blocks of understanding. 2 © Dr. Charles Smith, 2006 The author asserts and reserves all rights. However, this resource can be freely copied,

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Building-blocks of Building-blocks of understandingunderstanding

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© Dr. Charles Smith, 2006© Dr. Charles Smith, 2006

The author asserts and reserves all rights.The author asserts and reserves all rights.

However, this resource can be freely copied, adapted, and However, this resource can be freely copied, adapted, and used by teachers under the following conditions:used by teachers under the following conditions:

a.a. The source is acknowledged in your presentation The source is acknowledged in your presentation

(cite as: Smith, C. (2006), (cite as: Smith, C. (2006), Building-Blocks of Building-Blocks of Understanding,Understanding, in ‘Reflections on Teaching’, in ‘Reflections on Teaching’, www.economicsnetwork.ac.uk/showcase/classroom.htm);www.economicsnetwork.ac.uk/showcase/classroom.htm);

b. Feedback is sent to the author (see next slide)…b. Feedback is sent to the author (see next slide)…

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In return for suspending copyright, the author would be In return for suspending copyright, the author would be grateful for the following feedback from users: grateful for the following feedback from users:

1.1. Your name, institution, e-mail addressYour name, institution, e-mail address2.2. Date of use of resourceDate of use of resource3.3. Type and level of class Type and level of class 4.4. Any amendments incorporated (you could attach Any amendments incorporated (you could attach

a copy of your presentation)a copy of your presentation)5.5. A brief evaluation of the usefulness of this A brief evaluation of the usefulness of this

resourceresource

Please e-mail to: [email protected] e-mail to: [email protected]

It is hoped to use feedback as a basis for a future It is hoped to use feedback as a basis for a future study; therefore please state if you would prefer your study; therefore please state if you would prefer your feedback to be quoted anonymously. feedback to be quoted anonymously.

Thanks.Thanks.

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Building-blocks of Building-blocks of understandingunderstanding

Dr Charles Smith Dr Charles Smith Swansea School of Education Swansea School of Education

April 2006April 2006

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A resource for A resource for introducing introducing students to students to average and average and

marginal valuesmarginal values

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INTRODUCTIONINTRODUCTION

Before embarking on study of business economics Before embarking on study of business economics or the theory of the firm, students need a thorough or the theory of the firm, students need a thorough grasp of the differences and relationships between grasp of the differences and relationships between AVERAGE and MARGINAL values. AVERAGE and MARGINAL values.

This resource enables students to INVESTIGATE This resource enables students to INVESTIGATE these differences and relationships in an accessible these differences and relationships in an accessible and tactile way.and tactile way.

Thus, an aspect of economics which students Thus, an aspect of economics which students typically find very abstract, theoretical and difficult, typically find very abstract, theoretical and difficult, becomes practical, kinesthetic, and easier to becomes practical, kinesthetic, and easier to understand.understand.

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Methodology

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EQUIPMENT REQUIRED:EQUIPMENT REQUIRED:

1.1. Plastic building blocks, e.g. the LEGO Plastic building blocks, e.g. the LEGO ® brand® brand

2.2. Squared paperSquared paper

3.3. (Optional) Microsoft Excel ® spreadsheet(Optional) Microsoft Excel ® spreadsheet

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STEP 1STEP 1

1.1. Split your class Split your class into groups of 3 or into groups of 3 or 4 students per 4 students per tabletable

2.2. Put plenty of Put plenty of blocks on each blocks on each tabletable

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STEP 2STEP 2

Ask each group to Ask each group to make towers make towers containing containing different (random) different (random) numbers of blocksnumbers of blocks

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STEP 3STEP 3

1.1. Place a ‘base’ on Place a ‘base’ on the table.the table.

2.2. Students are going Students are going to add towers to to add towers to the base in the base in ASCENDING order ASCENDING order of height.of height.

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STEP 4STEP 4

aa bb cc dd ee

Tower Tower no.no.

Height of Height of this this tower (in tower (in blocks)blocks)

Total Total height so height so far far (blocks)(blocks)

Average Average height height (blocks per (blocks per tower)tower)

Marginal Marginal height height (blocks)(blocks)

11 22 22 2.02.0 22

This is tower no. 1; its This is tower no. 1; its height is 2 blocks. The height is 2 blocks. The average height of towers on average height of towers on the base is 2/1 = 2 blocks the base is 2/1 = 2 blocks per tower. The marginal per tower. The marginal height (blocks added to the height (blocks added to the base by the last tower) is 2.base by the last tower) is 2.

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STEP 5STEP 5

aa bb cc dd ee

Tower Tower no.no.


Total Total height so height so farfar


Marginal Marginal height (in height (in blocks)blocks)

11 22 22 2.02.0 22

22 55 77 3.53.5 55

Tower number 2 is added to Tower number 2 is added to the base, and the the base, and the corresponding numbers corresponding numbers inserted in the table.inserted in the table.

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STEP 5 (Continued)STEP 5 (Continued)

aa bb cc dd ee

Tower Tower no.no.





11 22 22 2.02.0 22

22 55 77 3.53.5 55

Two important points arise Two important points arise here. Firstly: marginal here. Firstly: marginal height can be calculated in height can be calculated in two ways: 1. the no. of two ways: 1. the no. of blocks in the latest tower on blocks in the latest tower on the base; 2. (total height) the base; 2. (total height) minusminus (previous total (previous total height)height)

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STEP 5 (Continued)STEP 5 (Continued)

aa bb cc dd ee

Tower Tower no.no.





11 22 22 22 2.02.0

22 55 77 55 3.53.5

The second point is that The second point is that average height (arithmetic average height (arithmetic mean) is less ‘real’ than mean) is less ‘real’ than marginal height. There is marginal height. There is nono tower on the base with 3.5 tower on the base with 3.5 blocks; there blocks; there isis one with 5 one with 5 blocksblocks

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STEP 6STEP 6

Continue by adding Continue by adding the next tower and the next tower and completing the tablecompleting the table

aa bb cc dd ee

11 22 22 2.02.0 22

22 55 77 3.53.5 55

33 77 1414 4.64.6 77

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STEP 7STEP 7

aa bb cc dd ee

11 22 22 2.02.0 22

22 55 77 3.53.5 55

33 77 1414 4.64.6 77

44 88 2222 5.55.5 88


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STEP 8STEP 8

aa bb cc dd ee

11 22 22 2.02.0 22

22 55 77 3.53.5 55

33 77 1414 4.64.6 77

44 88 2222 5.55.5 88

55 99 3131 6.26.2 99


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INVESTIGATIONSINVESTIGATIONS

Students can now use this table to draw a Students can now use this table to draw a graph on squared paper showing average graph on squared paper showing average height and marginal height on the Y axis, height and marginal height on the Y axis, and no. of towers on the X axis.and no. of towers on the X axis.

Then they can experiment with different Then they can experiment with different sequences of towers, and plot graphs to sequences of towers, and plot graphs to investigate relationships between average investigate relationships between average values (AV) and marginal values (MV).values (AV) and marginal values (MV).

THEY CAN THEN TRY TO DERIVE A THEY CAN THEN TRY TO DERIVE A LIST OF ‘RULES’ GOVERNING AV LIST OF ‘RULES’ GOVERNING AV

AND MVAND MV

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RULE 1RULE 1

Where AV is rising, MV is Where AV is rising, MV is ABOVE AVABOVE AV

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0123456789

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1 2 3 4 5

Marginal value

Average value

TowersTowers

BlocksBlocks(Rule 1)(Rule 1)

Tower Tower heightsheights

22

55

77

88

99

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RULE 2RULE 2

Where AV is falling, MV is Where AV is falling, MV is BELOW AVBELOW AV

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0123456789

10

1 2 3 4 5

Marginal value

Average value

TowersTowers



99

88

77

55

22

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RULE 3RULE 3

It follows logically from Rules It follows logically from Rules 2 and 1 that as it ascends, the 2 and 1 that as it ascends, the MV graph MV graph mustmust pass through pass through the MINIMUM point of the AV the MINIMUM point of the AV graph. So if AV graph. So if AV fallsfalls and then and then risesrises the graphs will look like the graphs will look like this…this…

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0123456789

10

1 2 3 4 5

Marginal value

Average value

TowersTowers



99

77

22

55

88

26

0

2

4

6

8

10

1 2 3 4 5

Marginal value

Average value

TowersTowers

BlocksBlocks

(Rule 3)(Rule 3)

NOTE: Eagle-eyed students might notice that in NOTE: Eagle-eyed students might notice that in this graph, the MV line does not this graph, the MV line does not quitequite pass pass through the minimum point of AV. This is because through the minimum point of AV. This is because the computer has plotted a straight line graph the computer has plotted a straight line graph instead of a smooth curve. You can reassure instead of a smooth curve. You can reassure students that Rule 3 is quite correct students that Rule 3 is quite correct mathematically.mathematically.

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RULE 4RULE 4

It follows logically from Rules It follows logically from Rules 1 and 2 that as it descends, the 1 and 2 that as it descends, the MV graph MV graph mustmust pass through pass through the MAXIMUM point of the AV the MAXIMUM point of the AV graph. So if AV graph. So if AV risesrises and then and then fallsfalls the graphs will look like the graphs will look like this…this…

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0123456789

10

1 2 3 4 5

Marginal value

Average value

TowersTowers



22

99

88

77

55

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RULE 5RULE 5

If AV is constant, MV If AV is constant, MV will also be constant, will also be constant,

and AV = MVand AV = MV

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0

1

2

3

4

5

6

1 2 3 4 5

Marginal value

Average value

TowersTowers



55

55

55

55

55

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Students could experiment with AV Students could experiment with AV and MV values using spreadsheets. and MV values using spreadsheets. The web-page article accompanying The web-page article accompanying this presentation includes this presentation includes spreadsheets with suitable formulae spreadsheets with suitable formulae already inserted. Students need only already inserted. Students need only add tower heights to column ‘b’ and add tower heights to column ‘b’ and the formulae will do the rest.the formulae will do the rest.

The shapes of the graphs will change The shapes of the graphs will change before their very eyes!before their very eyes!

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Having experimented with Having experimented with plastic blocks, and plastic blocks, and

graphs, and/ or graphs, and/ or spreadsheets, your spreadsheets, your

students should find that students should find that diagrams like those on the diagrams like those on the

following slides hold no following slides hold no horrors for them…horrors for them…

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OUTPUT

UNITS OF VARIABLE FACTOR

O

Marginal physical product

Average physical product

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MC

AC

COST

OUTPUT O

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REVENUE

OUTPUT O

MR AR

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REVENUE,COST

OUTPUT O

MR AR

MC

AC

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REVENUE

OUTPUT O

MR = AR

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REVENUE, COST

OUTPUT O

MR = AR

MC

AC

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ENDEND

1 Building-blocks of understanding. 2 © Dr. Charles Smith, 2006 The author asserts and reserves all rights. However, this resource can be freely copied,

Documents

slide slide

blocks average height

tower marginal height

blocks total height

av slide

mv slide

average height of towers

understanding slide