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Exploring Triangular Numbers Through Pascal’s Triangle and Party Handshakes

This investigation delves into the fascinating world of mathematical modeling using Pascal’s Triangle. We will generate the first eight rows of Pascal's Triangle, identifying linear, rectangular, and triangular numbers within its diagonal rows. By modeling a party handshake scenario with points and line segments, we will explore patterns in our data, noting that differences increase incrementally. Triangular numbers, which can be visualized as triangular arrangements, are examined alongside their links to rectangular numbers. This study highlights the relationship and significance of these numerical patterns.

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Exploring Triangular Numbers Through Pascal’s Triangle and Party Handshakes

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  1. Section 2-3 Mathematical Modeling – Triangular Numbers HW: Investigate Pascal’s Triangle Write out the first 8 rows Identify the linear, rectangular & triangular numbers in the diagonal rows

  2. 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.

  3. 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.

  4. 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 10 6 15

  5. 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.

  6. 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?

  7. 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.

  8. Rectangular Numbers • Rectangular number can be visualized as rectangular arrangement of objects.

  9. 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.

  10. 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.

  11. If the nth term for the rectangular numbers is n(n+1) • Then the nth term for the triangular numbers is 2 6 12 20

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