CReaTE 1: Critical Thinking Exploration 1
Thinking Inductively about Adding Positive Integers
Prepared by Shari Davis and Katherine Smith as part of a series of supplemental critical thinking explorations to promote discovery and deeper comprehension.
This project was supported by a 2017 Faculty Innovator Grant through Old Dominion University’s Center for Learning and Teaching.
Part 1
Getting Ready
The first part of this exploration introduces a motivating example. Students should have some familiarity with inductive reasoning. They are expected to have familiarity with the terminology below:
Inductive reasoning is the process of reasoning from specific examples to a general conclusion. We can think of inductive reasoning as being similar to experimentation. We review lots of evidence and then draw a single conclusion. The conclusion we draw is called a conjecture or hypothesis. We can think of our conjecture as an educated guess. Since inductive reasoning is based on evidence and experience, our conjecture could be incorrect. All we need is a single example, called a counterexample, to contradict our conjecture in order to show that the result of our inductive reasoning was incorrect.
Video
Students should watch the following video: Critical Thinking Exploration 1 Part 1 Video
Follow-up
After watching the video, students should be given time to think about and discuss the next line in the example. This could take place in class, after class or using an online discussion board.
A sample discussion board prompt would be: What should the next line in the example be? Can you come up with the line after that?
Part 2
Getting Ready
Students should have viewed Part 1 of the series and had time to complete the follow-up activity.
Video
During this activity, students will be led to discover the formula for adding the first N positive integers.
Students should watch the following video: Critical Thinking Exploration 1 Part 2 Video
Follow-up
After watching the video, students should be given time to think about and discuss whether or not they can always cancel the 2 in the formula. This could take place in class, after class or using an online discussion board.
A sample discussion board prompt would be: Do you think you can always cancel the 2 in the formula we discovered for adding the first N positive integers? Why or why not?
Part 3
Getting Ready
Students should have viewed Part 2 of the series and had time to complete the follow-up activity.
Video
During this activity, students will be introduced to a practical example where the presenter used the formula for adding consecutive integers.
Students should watch the following video: Critical Thinking Exploration 1 Part 3 Video
Follow-up
After watching the video, students should be given time to think about and discuss whether they have ever used inductive reasoning in their everyday life. This could take place in class, after class or using an online discussion board.
A sample discussion board prompt would be: Have you ever used inductive reasoning in your everyday life?
It may be helpful to provide some examples to stimulate the discussion. Here are a few examples
- Every time you leave for work after 8 am you are late. You left at 8:15 am this morning and you assume that you will be late for work.
- You watch the first few episodes of a new television show and really enjoy it. You assume that you will continue to enjoy the show and make time to watch it every week.
Part 4
Getting Ready
Students should have viewed Part 3 of the series and had time to complete the follow-up activity.
Video
During this activity, students will be led to modify the formula for adding the first N positive integers.
Students should watch the following video: Critical Thinking Exploration 1 Part 4 Video
Follow-up
After watching the video, students should be given time to think about and discuss how the new example differs from the previous examples and how we might adapt what we are learning to solve this new problem. This could take place in class, after class or using an online discussion board.
A sample discussion board prompt would be: How does the new example differ from the previous examples? Do you have any ideas on how to use our formula to calculate the new sum?
Part 5
Getting Ready
Students should have viewed Part 4 of the series and had time to attempt the follow-up activity. Alternatively, this video can be viewed directly after Part 4 as it provides a hint to help students with the follow-up activity.
Video
During this activity, students will be given a hint to help them complete the follow-up activity from Part 4.
Students should watch the following video: Critical Thinking Exploration 1 Part 5 Video
Follow-up
After watching the video, students should be given time to continue to think about and discuss the follow-up activity from Part 4. This could take place in class, after class or using an online discussion board.
A sample discussion board prompt would be: Now that you have been shown an additional hint, can you determine how the new example differ from the previous examples? Do you have any ideas on how to use our formula to calculate the new sum?
Part 6
Getting Ready
Students should have viewed Part 5 of the series and had time to complete the follow-up activity.
Video
During this activity, students will be presented with the solution to the follow-up activity from Parts 4 and 5.
Students should watch the following video: Critical Thinking Exploration 1 Part 6 Video
Follow-up
After watching the video, students should be given time to come up with the right-hand-side for the two new examples. They can evaluate them numerically to get a final answer if they want. This could take place in class, after class or using an online discussion board.
A sample discussion board prompt would be: How would you find the sums for the two examples presented at the end of the video?
Part 7
Getting Ready
Students should have viewed Part 5 of the series and had time to complete the follow-up activity.
Video
During this activity, students will be presented with the solution to the follow-up activity from Part 6.
Students should watch the following video: Critical Thinking Exploration 1 Part 7 Video
Follow-up
This video concludes Part 1. After watching the video, students could discuss other problems they might be able to solve now. This could take place in class, after class or using an online discussion board.
Examples of other problems that can be solved similarly are:
- Finding the sum of odd positive integers.
- Finding sums when negative integers are included.
- Finding the sum of every fifth integer.
