Transcript Section 2-1

SECTION 2-3
Mathematical Modeling – Triangular Numbers
HW: Investigate Pascal’s Triangle
o Write out the first 8 rows
o Identify the linear, rectangular & triangular
numbers in the diagonal rows
OBJECTIVES
• When you draw graphs or pictures of situation or when
you write equations that describe a problem, you are
creating a mathematical model.
• A physical model of a complicated telecommunication
network, for example, might not be practical, but you can
draw a mathematical model of the network using points
and lines.
Party Handshake
• Model the problem by using points to represent people
and line segments connecting the points to represent
handshakes.
• Record your results in a table.
Party Handshake
• Look at your table. What patterns do you see?
• The pattern of differences is increasing by one: 1, 2, 3, 4,
5, 6, 7, etc. The pattern of differences is not constant as
we have seen in other tables. So this pattern is not linear.
3
6
10
15
Party Handshake
• In the diagram with 5 vertices, how many segments are
there from each vertex? So the total number of segments
written in factored form is
• Complete the table below by expressing the total number
of segments in factored form.
Party Handshake
• The larger of the two factors in the numerator represent
the number of points. What does the smaller of the two
numbers in the numerator represent? Why do we divide
by 2?
• Write a function rule. How many handshakes were there
at the party?
Triangular Numbers
• The numbers in the pattern in the previous investigation
are called triangular numbers because you can arrange
them into a triangular pattern of dots.
• The triangular numbers appear in numerous geometric
situations.
• The triangular numbers are related to the sequence of
rectangular numbers.
Rectangular Numbers
•
Rectangular number can be visualized as rectangular
arrangement of objects.
•
•
•
•
In the sequence the width is equal to the term number
and the length is one more than the term number.
The third term:
• the rectangle is 3 wide and the length is 3+1 or 4.
The fourth term:
• the rectangle is 4 wide and the length is 4+1 or 5.
The nth term:
• the rectangle is n wide and the length is n+1.
Relating the rectangle and the triangle
• Here is how the triangular and rectangular numbers are
related:
2
6
12
20
• The triangular numbers are ½ of the rectangular numbers.
• If the nth term for the rectangular numbers is n(n+1)
• Then the nth term for the triangular numbers is n(n  1)
2
2
6
12
20