FFPC+Mathematical+Reflections,+p+51

Feb. 8, 2009 NG //Big Idea//: Quadratic functions can help us describe situations in the real world. //Essential Question//: What does a quadratic function look like in a situation, table, graph, or equation? //Notes From Class//: **triangular numbers**- quantities that can be arranged in a triangular pattern

The handshake problems and dot-pattern problems are similar in that they are both quadratic relationships, so when you graph them a parabola will form. Additionally, they both increase by increasing numbers. Their equations are also very similar: 3.1 (handshake) y= (n(n-1))/2 3.2 (dot-pattern) y= (n(n+1))/2 The handshake problems and dot-pattern problems are different in that the dot-pattern values continue increasing, while the handshake values equal a total number of hanshakes that are possible.
 * 1. In what ways are the relationships in the handshake problems similar to the relationships in the dot-pattern problems? In what ways are these relationships different?**

In the 1st investigation, the quadratic functions, when graphed, appeared mostly in the upper right-hand quadrant (only positive values) of the graph. In the 2nd and 3rd investigations, the parabolas stretched to many quadrants (positive and negative values). Also, the equations were simpler in Investigation 1: y=(x-m) than in Investigation 2: y=x(x-m) or y=x^2-mx or Investigation 3: y=(x(x+m))/z However, all of the investigations' functions make parabolas, because they're quadratic functions, and have a maximum point.
 * 2. In what ways are the quadratic functions in this investigation similar to the quadratic functions in Investigation 1 and 2? In what ways are they different?**

For the handshake problems, a constant change in the x-values for these functions doesn't change the y-values. In the dot-pattern problems, the pattern of change is each value increases by consecutive values. If it is a problem like case 1 in 3.1, then you know the next entry is exactly the same as the previous entries. Otherwise, you can see how much every entry is increasing, and use that pattern to predict the next one.
 * 3.a. Describe the patterns of change for the quadratic functions in this investigation.**
 * b. In a table of values for a quadratic function, how can you use the pattern of change to predict the next entry?**

In this investigation, I learned about quadratic patterns of change. I learned how these patterns can be applied to real-life situations, like in the handshake/high five problem. I also found out what a triangular number is, and what numbers can be called triangular. Most importantly, I learned that even though 3.1 and 3.2 dealt with different forms of quadratic relationships, their patterns of change and graphs were quite similar.
 * __SUMMARY__**