On the back of this page are scale drawings of windows and their prices. The manufacturer bases the suggested price on the lineal feet of wood for the frame, the number of panes of glass, the square footage of the glass and a fixed overhead fee, all of which are whole dollar amounts. You are to determine the formula the manufacturer actually uses to determine the suggested prices. You need to support all your conclusions with the applicable mathematics and clear explanations. There is one incorrectly priced window that you will need to correct.
The Scoring:
You will be scored on a four-point scale for each bulleted area:
• Show understanding of scale drawings
• Show ability to correctly apply algorithms
• Support conclusions with applicable mathematics and clear explanations
• Collect and represent data in an organized fashion
• Represent the mathematical processes and ideas in an effective format
• Show understanding of the problem
• Correctly complete all of the required tasks
| A $234 | B $148.50 | C $183 | D $119.50 | E $200 | F $284 | G $347 | H $315 | I $189.50 | J $147 | K $245 | L $275 | M $405.50 | N $153.50 |
(Measure wood and glass from the middle of the heavy black line to the middle of the heavy black line and round those lengths to the nearest quarter foot as shown above and to the right.)
The Scoring:
You will score teams papers on a four-point scale for each bulleted area:
4 - meets or exceeds all relevant criteria
3 - meets most relevant criteria
2 - meets some relevant criteria
1 - meets few relevant criteria
0 - not scoreable: off topic, no attempt, can't be read, etc.
The Scoring:
You will be scored on a four-point scale for each bulleted area:
• Show understanding of scale drawings
The team clearly demonstrates the ability to measure and to convert measurements from centimeters to feet for the area of the glass and the lineal footage of the wood frame. Did they get reasonable values for area, etc.
• Show ability to correctly apply algorithms
The team clearly demonstrates the ability to correctly apply algorithms. For example if they wrote systems of equations, did they correctly apply algorithms to solve them?
• Support conclusions with applicable mathematics and clear explanations
The team shows evidence that they made connections between the data and the mathematical processes that are helpful in solving the problem. For example did they deduce that the formula is a four variable function or that the problem might be solved by isolating a single variable.
• Collect and represent data in an organized fashion
The team will most likely make some sort of a table or chart of the data to organize it.
• Represent the mathematical processes and ideas in an effective format
How well did the team show or explain their use of mathematics to arrive at their conclusions.
• Show understanding of the problem
The team showed correct interpretation of the problem and chose a reasonable approach for solving the problem.
• Correctly complete all of the required tasks
The team found: cost per sq. ft. of the glass, cost per lineal ft. of the wood, cost for multiple panes, the fixed cost, and corrected the price of the one window.
Sample Solution:
| Window | Area A | Wood W | Panes P | Marked Costs | Checked Costs |
| A | 14 | 15 | 1 | 234 | 234 |
| B | 6.125 | 10.5 | 1 | 148.5 | 148.5 |
| C | 8.25 | 14 | 1 | 183 | 183 |
| D | 3.75 | 8.5 | 1 | 119.5 | 119.5 |
| E | 6.75 | 15 | 3 | 200 | 200 |
| F | 12 | 21 | 4 | 284 | 284 |
| G | 25 | 20 | 1 | 347 | 347 |
| H | 21 | 20 | 1 | 315 | 315 |
| I | 5.625 | 11.5 | 1 | 189.5 | 149.5 |
| J | 6.25 | 10 | 1 | 147 | 147 |
| K | 9 | 18 | 4 | 245 | 245 |
| L | 16 | 20 | 1 | 275 | 275 |
| M | 22.5 | 28.5 | 4 | 405.5 | 405.5 |
| N | 6.75 | 10.5 | 1 | 153.5 | 153.5 |
Because G, H, and L all have 20 lineal feet of wood, one pane, and the fixed cost, the difference in their prices must be due to the difference in areas. Between G and H is $32 and 4 sq. ft. or $8 per sq. ft. Between H and L is $40 and 5 sq. ft. or $8 per sq. ft. Between G and L is $72 and 9 sq. ft. or $8 per sq. ft. Since there is only one wrong price the cost for the area per sq. ft. must be $8.
Using L and N, multiplying the areas by $8 and subtracting that from the total costs would leave the price for the difference in the wood since the number of panes is equal and the fixed cost is also the same. For L: $275-$8(16)=$147. For N: $153.5-$8(6.75)=$99.50. The difference is $47.50 and that divided by difference in the wood (9.5 ft.) gives $5 per lineal ft. of wood. The same value occurs when the same procedure is applied to A and C so the cost per lineal foot of wood frame must be $5.
Applying these two values and similar procedures to A and F or A and K, the value for the panes must be $12 per pane. From that and using A, $234-$8(14)-$5(15)-$12(1)=$35 which must be the fixed cost.
Therefore the cost must be calculated by the formula $8(A)+$5(W)+$12(P)+$35. Using this formula on the table to check the costs, we see that window I is incorrectly priced and should be $149.50.
Alternate approach:
This is a system with four variables fitting the equation in the form:
A(c1)+W(c2)+P(c3)+c4=Price.
Therefore using A, B, C, and E we get the system:
14(c1) + 15(c2) + 1(c3)+c4=234
6.125(c1) +10.5(c2) + 1(c3)+c4=148.5
8.25(c1) + 14(c2) + 1(c3)+c4=183
6.75(c1) + 15(c2) + 3(c3)+c4=119.5
Using matrices we get costs to be $8, $5, $12 and $35. We tried another set of 4, like E, F, G, and H and got the same results. (and then as above)
© 2000 WSMC, Jim Miller, and Richard Edgerton