Drawing a rectangle (using algebra)
Problem.
Draw a rectangle with perimeter 25 cm and with the ratio of its sides being 3/4.
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A model solution.
1. Sketch a rectangle,
2. Choose variables.
|
p |
perimeter; |
|
r |
ratio of sides; |
|
x, y |
lengths of sides. |
3. Write relations among variables.
|
p = x + y + x + y = 2x + 2y |
(1) |
|
r = y/x |
(2) |
4. Find x and y.
|
y = r*x |
(3) from (2) |
|
p = 2x+ 2r*x = (2 + 2r)x |
(4) from (1) and (3) |
|
x = p/(2 + 2r) |
(5) from (4) |
|
y = r*x = r*p/(2 + 2r) |
(6) from (5) |
5. Compute the numerical values of x and y.
Compute mentally,
x = 25/3.5
y = 0.75*x
| Keystrokes: | Display: | Write down: |
| [25][/][3.5][=] | 7.1428571 | x = 7.1 cm |
| [*][.75][=] | 5.3571429 | y = 5.4 cm |
6. Check your answers.
Compute mentally,
(7.1 + 5.2)*2 = 25
| Keystrokes: | Display: |
| [5.4][/][7.1][=] | 0.7605634 |
7. Draw the required rectangle.
Use a ruler with centimeters and millimeters, and a protractor or index card to draw right angles. You may use also a compass.
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Comments.
1. Using the multiplication sign, *, is optional. But using it between any two variables is a good habit.
2. The calculator used here was the TI-30. Finishing algebraic problems with numerical values is important. It shows the use of formulas as a basis for programs, it catches errors, and it provides a tangible purpose for algebraic manipulations.
3. Do not force children to memorize formulas. With experience in arithmetic and clear meanings of the variables, they will "see" that some equalities are true, and in this way they will develop skills in mental algebra, similar to skills in mental arithmetic.
4. Keeping track of which equality follows from which is important. Without doing this, it is very difficult to follow someone else’s reasoning, and even to follow one's own reasoning after a day or two.