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4 Building-blocks of understanding Dr Charles Smith Swansea School of Education April 2006
5 A resource for introducing students to average and marginal values
6 INTRODUCTION Before embarking on study of business economics or the theory of the firm, students need a thorough grasp of the differences and relationships between AVERAGE and MARGINAL values. This resource enables students to INVESTIGATE these differences and relationships in an accessible and tactile way. Thus, an aspect of economics which students typically find very abstract, theoretical and difficult, becomes practical, kinesthetic, and easier to understand.
8 EQUIPMENT REQUIRED: 1.Plastic building blocks, e.g. the LEGO ® brand 2.Squared paper 3.(Optional) Microsoft Excel ® spreadsheet
9 STEP 1 1.Split your class into groups of 3 or 4 students per table 2.Put plenty of blocks on each table
10 STEP 2 Ask each group to make towers containing different (random) numbers of blocks
11 STEP 3 1.Place a base on the table. 2.Students are going to add towers to the base in ASCENDING order of height.
12 STEP 4 abcde Tower no. Height of this tower (in blocks) Total height so far (blocks) Average height (blocks per tower) Marginal height (blocks) This is tower no. 1; its height is 2 blocks. The average height of towers on the base is 2/1 = 2 blocks per tower. The marginal height (blocks added to the base by the last tower) is 2.
13 STEP 5 abcde Tower no. Height of this tower (in blocks) Total height so far Average height (blocks per tower) Marginal height (in blocks) Tower number 2 is added to the base, and the corresponding numbers inserted in the table.
14 STEP 5 (Continued) abcde Tower no. Height of this tower (in blocks) Total height so far Average height (blocks per tower) Marginal height (in blocks) Two important points arise here. Firstly: marginal height can be calculated in two ways: 1. the no. of blocks in the latest tower on the base; 2. (total height) minus (previous total height)
15 STEP 5 (Continued) abcde Tower no. Height of this tower (in blocks) Total height so far Marginal height (in blocks) Average height (blocks per tower) The second point is that average height (arithmetic mean) is less real than marginal height. There is no tower on the base with 3.5 blocks; there is one with 5 blocks
16 STEP 6 Continue by adding the next tower and completing the table abcde
17 STEP 7 abcde Continue by adding the next tower and completing the table
18 STEP 8 abcde Continue by adding the next tower and completing the table
19 INVESTIGATIONS Students can now use this table to draw a graph on squared paper showing average height and marginal height on the Y axis, and no. of towers on the X axis. Then they can experiment with different sequences of towers, and plot graphs to investigate relationships between average values (AV) and marginal values (MV). THEY CAN THEN TRY TO DERIVE A LIST OF RULES GOVERNING AV AND MV
20 RULE 1 Where AV is rising, MV is ABOVE AV
21 Towers Blocks (Rule 1) Tower heights
22 RULE 2 Where AV is falling, MV is BELOW AV
23 Towers Blocks (Rule 2) Tower heights
24 RULE 3 It follows logically from Rules 2 and 1 that as it ascends, the MV graph must pass through the MINIMUM point of the AV graph. So if AV falls and then rises the graphs will look like this…
25 Towers Blocks (Rule 3) Tower heights
26 Towers Blocks (Rule 3) NOTE: Eagle-eyed students might notice that in this graph, the MV line does not quite pass through the minimum point of AV. This is because the computer has plotted a straight line graph instead of a smooth curve. You can reassure students that Rule 3 is quite correct mathematically.
27 RULE 4 It follows logically from Rules 1 and 2 that as it descends, the MV graph must pass through the MAXIMUM point of the AV graph. So if AV rises and then falls the graphs will look like this…
28 Towers Blocks (Rule 4) Tower heights
29 RULE 5 If AV is constant, MV will also be constant, and AV = MV
30 Towers Blocks (Rule 5) Tower heights
31 Students could experiment with AV and MV values using spreadsheets. The web-page article accompanying this presentation includes spreadsheets with suitable formulae already inserted. Students need only add tower heights to column b and the formulae will do the rest. The shapes of the graphs will change before their very eyes!
32 Having experimented with plastic blocks, and graphs, and/ or spreadsheets, your students should find that diagrams like those on the following slides hold no horrors for them…
33 OUTPUT UNITS OF VARIABLE FACTOR O Marginal physical product Average physical product