2.1+Tiling+Pools+0809

April 4, 2009 PR Notes:  4s+4 = 4(s+1) = s x 4 + 4 = (s+2) x (s+2) + 52

In this problem, you will explore this question: If a square pool has sides of length s feet, how many tiles are needed to form the border? Record your results in a table. Side Length || Border Total || 1 || 8 || 2 || 12 || 3 || 16 || 4 || 20 || 6 || 28 || 10 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">43 || <span style="color: black; font-family: 'Arial','sans-serif'; mso-bidi-font-size: 12.0pt; mso-fareast-font-family: 'Times New Roman';"> N = 4(s+1)
 * A**. Make sketches on grid paper to help you figure out how many tiles are needed for the border of square pools with sides of length 1, 2, 3, 4, 6 and 10 feet.
 * B**. Write an equation for the number of tiles, N, needed to form a border for a square pool with sides of length s feet.

N = 4s + 4 To convince someone that they are equivalent you can use the distributive property that proves that the two equations, N = 4(s+1) and N = 4s + 4, are equivalent.
 * C**. Try to write at least one more equation for the number of tiles needed for the border of the pool. How could you convince someone that your expressions for the number of tiles are equivalent?

1. Make a table and a graph for each equation you wrote in part A, of Problem 2.1 Do the table and graph indicate that the equations are equivalent? <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">Side Length || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">n = 4(s+1) || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">n = 4s + 4 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">1 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">8 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">8 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">2 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">12 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">12 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">3 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">16 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">16 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">4 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">20 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">20 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">5 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">24 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">24 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">6 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">28 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">28 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">7 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">32 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">62 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">8 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">36 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">36 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">9 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">40 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">40 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">10 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">44 || <span style="font-size: 10pt; color: black; font-family: 'Arial','sans-serif'; mso-fareast-font-family: 'Times New Roman';">44 || <span style="color: black; font-family: 'Arial','sans-serif'; mso-bidi-font-size: 12.0pt; mso-fareast-font-family: 'Times New Roman';"> Graph that compares the two equations: The table shows that they are equal because they both show the same data. And on the graph, both the equations make the exact same line. So this means that they are equivalent.
 * FOLLOW UP**-
 * 2**. Is the relationship between the side length of the pool and the number of tiles linear, quadratic, exponential, or none of these? Explain your reasoning.

I think that the relationship is linear, because if you look on the graph it makes a straight line. Also, you can see that as x goes up by 1, y goes up by 4. A = s² b. Is this equation you wrote linear, quadratic, exponential, or none of these? Explain. The equation is quadratic, because it forms the shape of a square. Also it has a power of 2, s².
 * 3**. a. Write an equation for the area of the pool, A, in terms of the side length, s

C= 3^2 + 4s + 4 or C = (s+2)^2 b. Is the equation you wrote linear, quadratic, exponential, or none of these? Explain. I think it is a quadratic relationship because it has the basic form of one, and also it has an exponent (power of two).
 * 4**.a. Write an equation for the combined area of the pool and its border, C, in terms of the side length, s.