3.2+Exploring+Triangular+Numbers


 * YMJ

Problem 3.2 Exploring Triangular Numbers

Note:

The numbers of dots in the figures above are called triangular numbers. The first triangular number is 1, the second triangular number is 3, third is 6, the fourth is 10, and so on.

What two variables are important in this situation? Which is the independent variable, and which is the dependent variable?

A.** The independent is figure. The dependent is number of dots.

Look for a pattern in the figure above. Use the pattern to help you make a table of the first ten triangular numbers.


 * B.**
 * **x** || **y** ||
 * 0 || 0 ||
 * 1 || 1 ||
 * 2 || 3 ||
 * 3 || 6 ||
 * 4 || 9 ||
 * 5 || 12 ||
 * 6 || 15 ||
 * 7 || 18 ||
 * 8 || 21 ||
 * 9 || 24 ||
 * 10 || 27 ||
 * Describe the pattern of change from one triangular number to the next.

C.

How can you use this pattern of change to predict the 15th triangular number without making a drawing?

D.** n+(n-1)


 * Write an equation that can be used to determine the nth triangular number.

E.

Does your equation represent a quadratic relationship? Explain your answer.

F.

Problem 3.2 Follow-up

1. Give the dimensions and the area of the rectangle formed by combining two copies of the given figure

a. the 4th figure

b. the 5th figure

c. the 10th figure

d. the nth figure

2. For each part of question 1, how does the number of squares in the rectangle compare with the number of squares in the original figure?

3. Use**