SAT to POW
In the SAT to POW project, we had to find an SAT problem, explore the mathematical concept behind it, and create an open-ended "POW" (Problem of the Week) based off of it. You can either see the final POW (and try to solve it) at the bottom of this page, or see how I got to that product first, by reading on. Be warned, though. The problem will be spoiled if you read on.
My thought process in creating this POW went in a relatively straight-forward path. Once I had found my problem, I answered it, then began exploring different options. Here is the original problem, found in a practice SAT book made by CollegeBoard:
1, 2, 2, 3, 3, 3, 4, 4, 4, 4...
All positive integers appear in the sequence above, and each positive integer k appears in the sequence k times. In the sequence, each term after the first is greater than or equal to each of the terms before it. If the integer 12 first appears in the sequence as the nth term, what is the value of n?
Once I had gotten the answer, 67, I began looking for an equation that would apply to any number that could take 12's (or, as I called it, s's) place. I decided to create a rule that used summation notation, though I didn't know it yet. I knew that I needed to count up to the first instance of s, rather than the last, so I used s-1 in the equation, then added 1 back in at the end. So far, I had something like this:
1+2+3+4+5+6+... (s-1)+1
I remembered that, to count consecutive sums in the way that I was, I needed to use summation notation. Once I realized this, I changed my equation to this:
1, 2, 2, 3, 3, 3, 4, 4, 4, 4...
All positive integers appear in the sequence above, and each positive integer k appears in the sequence k times. In the sequence, each term after the first is greater than or equal to each of the terms before it. If the integer 12 first appears in the sequence as the nth term, what is the value of n?
Once I had gotten the answer, 67, I began looking for an equation that would apply to any number that could take 12's (or, as I called it, s's) place. I decided to create a rule that used summation notation, though I didn't know it yet. I knew that I needed to count up to the first instance of s, rather than the last, so I used s-1 in the equation, then added 1 back in at the end. So far, I had something like this:
1+2+3+4+5+6+... (s-1)+1
I remembered that, to count consecutive sums in the way that I was, I needed to use summation notation. Once I realized this, I changed my equation to this:
To account for the hypothetical zero-th case, though, I had to change my bottom variable to 0. That meant I finished with this:
Finally, I began to investigate cases where I removed all even or all odd numbers from the sequence. Sadly, though, I wasn't able to find much of a pattern in those cases within the time I had.
Sequence Says
Here is a sequence of numbers:
1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5...
Within this sequence, there are many patterns, and many possibilities for exploration. We are going to look at some of those patterns now!
Part 1: Look for patterns in the sequence.
Part 2: Explore the problem, find some more patterns, then find an equation
Extension:
Alter this sequence of numbers to include only even numbers/only odd numbers. How does that change your written rules? Your formula?
Write Up:
Original SAT problem:
Adapted from “The Official SAT Study Guide With DVD” by CollegeBoard. Page 530, problem 16. (New York, CollegeBoard, 2012).
1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5...
Within this sequence, there are many patterns, and many possibilities for exploration. We are going to look at some of those patterns now!
Part 1: Look for patterns in the sequence.
- What do you expect the next number in the sequence will be? After that? What about the following numbers?
- Are there any patterns in the sequence? Write them down and make them into “rules” (with words, not numbers).
Part 2: Explore the problem, find some more patterns, then find an equation
- If the number 12 first appears in the sequence as the nth term, what is the value of n? In other words, at what point does the number 12 first appear in the sequence?
- Do more problems like 1a.
- Make a generalization based on your findings.
- Create a formula based on your findings.
Extension:
Alter this sequence of numbers to include only even numbers/only odd numbers. How does that change your written rules? Your formula?
Write Up:
- Problem Statement - What was the problem about?
- Process - Describe your thinking process during this problem. What ideas worked? What ideas didn’t work? Where did you get stuck and how did you get unstuck?
- Solution - Give specific examples of your solution. If you figured out a formula/equation, prove that it works.
- Reflection - Reflect on this problem. Include two Habits of a Mathematician
Original SAT problem:
Adapted from “The Official SAT Study Guide With DVD” by CollegeBoard. Page 530, problem 16. (New York, CollegeBoard, 2012).