Rectangle Simulation
- Author:
- Tom Owsiak
Rectangle with Perimeter Constraint
Name:
Date:
1) Work slowly and carefully. Mathematics is the art of careful reasoning.
2) Ultimately, being careful the first time pays because it saves energy and spares unnecessary frustration.
3) Don’t hesitate to contact me if you have questions about anything contained in this packet.
4) Don’t forget to go online to use the demonstration. It will be helpful to you as you build a proper understanding of this fascinating subject.
5) Learning is a process that occurs over time, so work on this packet over an extended period of time.
Exploration 2:
It is possible to get useful information from an equation or system of equations without finding the values of the variables that satisfy the equation or system of equations. Below is a rectangle. Answer the questions that follow.
1) Label the unlabeled sides with the correct variables.
2) Write and simplify an expression to represent the perimeter of the rectangle.
Perimeter=
3) Say it happens that the rectangle perimeter is equal to 16. Write an equation to represent this fact.
4) Now divide the equation in step 3) above by 2. Write a sentence to explain what this new equation represents. Also, in the picture below, trace over a portion of the rectangle represented by this new equation.
5) Now take the equation from step 3) above and divide it by 4. Write a sentence to explain the meaning of this equation. Here, notice that, because there are four sides, dividing by 4 amounts to finding what?
6) Let’s summarize so far. Do you know what x or y is individually in either question 3, 4 or 5) above?
7) Have you been able to discover new facts about our little rectangle despite the answer to question 6) above?
8) Now take the equation from step 3) above and divide both sides by 100. What does this equation represent? Mark a reasonable portion of the perimeter represented by this new equation.
9) Now using the x and y intercepts, graph the equations from steps 3), 4), 5) and 8) below. There are four coordinate planes, so keep them separate.
10)
11)
12)
13)
10) Are the graphs all the same, despite the different appearances of the equations?
11) Now I won’t tell you what to do, but how long is 1/8 of the perimeter of this rectangle? Modify both sides of the equation from step 3) above to find this value. Mark a reasonable portion of the perimeter in the picture.
12) Now look at the graphs again. Is the point a (3, 5) on all the lines?
13) So, then, according to question 12) above, can you conclude that, perhaps, in general, if a point satisfies one equation, it also satisfies any equation correctly derived from that equation?
14) Do you think there is anything special about any of the equations from the steps above? That is, is any equation more or less important than any other equation, or do you use the one that is more relevant in a given context?
15) Let’s investigate the variables we have here. This is within the context of the equation from step 3) above. Remember, this means that the perimeter is ALWAYS 16. Look at the picture below.
Picture 1.
Picture 2.
Picture 3.
Picture 4.
Can you vary y without varying x? For example, if y decreases, what has to happen to the value of x in order for the sum to remain at 8? (remember, x+y=8 is half the perimeter) Look at the pictures in numerical order.
16) Look at the pictures below. Every time x increases by 2, y decreases by what number? (think slope)
17) Look at the pictures below again. Every time x decreases by 2, y increases by what amount? (think slope)
18) Look at the pictures above again. Say somebody says that x can be 9. Is this possible within the context of having x+y=8? (remember, x and y are the sides of rectangles, so can they be negative? These are physical quantities you can measure with a ruler. Think of it this way: have you ever seen “-2” dogs being walked? )