I previously wrote about how I used (and am still using) GPT-4 to do a curriculum review. I did that review for my Algebra 2 course and I did the new curriculum plan on a new Statistics/Probability course. Unfortunately this means no more teaching Calculus :( but it was a good run. I love the subject, it's so interesting to me in how it describes real life, so it'll be missed but I might just put my lessons online. That way someone else could use them or learn from them, hmm...
In the meantime, I wanted to show you an example of how I would be reviewing and revising my Calculus presentation slides before teaching it again, if I were teaching it. Here's my first lesson of the year (after prerequisites) on introducing the concept of a limit: https://docs.google.com/presentation/d/1XwtgxMmjtSYJ59Z1C4STCjXu5PL-k7u4JEpnM69W4Vk/edit?usp=sharing
For the following examples, my prompts to GPT-4 with plugins Wolfram Alpha and LinkReader are bolded and GPT's answers are not.
To reimagine the presentation with the help of GPT-4, I first did this prompt:
Let's go through the presentation linked below together.
First, start by summarizing the key points from the presentation. After that, proceed through the presentation and summarize the main learnings from it. Finally, provide an enticing overall summary, make it compelling for the reader, and suggest ways to improve the presentation.
Second, critique your own summary. Does it accurately reflect the main points of the presentation? Is it compelling enough to make someone want to read the presentation? What improvements can be made to the summary?
Third, imagine how experts in math pedagogy and teaching, like Jo Boaler and Sal Khan, would summarize and critique this linked presentation. What key points would they focus on? What aspects of the presentation might they suggest to improve for better understanding?
Finally, synthesize all of those points above into one coherent and beautiful answer. Then provide your 3 biggest improvements you would make to either the flow of information or to the presentation to improve the student's understanding of the topic and include 3 examples of each of the improvements.
The linked presentation is:
with the Google Slides link I shared before. The results of that prompt was:
**Summary of the Presentation**
The presentation titled "Limits: Intro to Limits" by Mr. Jarrell on September 2, 2022, is an introductory lesson on limits in calculus. The presentation begins with a review of a homework assignment, followed by an exploration of the concept of a limit, one-sided limits, and the Limit Theorem. The presentation is interactive, with several 'You Do' sections where students are encouraged to find the limit by plugging in values. The presentation concludes with a list of resources for further study, including Khan Academy videos and other links.
**Critique of the Summary**
The summary provides a clear overview of the presentation's content, highlighting the main topics covered and the interactive nature of the lesson. However, it could be more compelling by emphasizing the real-world applications of limits and how understanding this concept is crucial in fields such as physics and engineering. The summary could also mention the use of multimedia resources, like the YouTube video, to make the lesson more engaging.
**Jo Boaler and Sal Khan's Perspective**
Jo Boaler, a renowned math educator, might appreciate the interactive nature of the presentation, as it aligns with her belief in active learning. She might suggest incorporating more visual representations of the concept of limits to cater to visual learners. Sal Khan, the founder of Khan Academy, might focus on the use of online resources and self-paced learning. He might suggest adding more step-by-step explanations and practice problems to enhance understanding.
**Synthesis and Improvements**
The presentation provides a comprehensive introduction to the concept of limits in calculus. It effectively uses interactive elements and online resources to facilitate understanding. However, to improve the flow of information and enhance student understanding, the following improvements could be made:
1. **Incorporate Real-World Examples**: To make the concept of limits more relatable and interesting, real-world examples where limits are used could be included. For instance, an example could be how limits are used in physics to calculate instantaneous velocity.
2. **Visual Representations**: Graphical illustrations of one-sided and two-sided limits could be included to cater to visual learners. This could involve showing how the limit changes as we approach from the left or right side.
3. **Step-by-Step Explanations**: More detailed, step-by-step explanations of how to calculate limits could be beneficial. This could involve breaking down the process into smaller, more manageable steps.
Overall, the presentation is a solid introduction to limits, but with these improvements, it could be even more effective in facilitating student understanding.
Then I wanted to see what improvements it would make. I then asked it this:
You are an AI program that is designed to create the world’s most fun, interesting, and informative math lessons for high schoolers. The lessons are not in the exact style of but are similar to Richard Feynman’s famous physics lectures. Above all, these lessons should be relatable to the students' lives so that the students care more about them.
Give me at least 20 specific examples or math problems that would make that linked presentation you just looked at the world’s best math lesson
and it spit out this:
Absolutely, I'd be happy to help make this lesson on limits more engaging and relatable to high school students. Here are 20 examples and problems that could be incorporated:
1. **The Speeding Car**: Imagine you're driving a car that accelerates from 0 to 60 mph in 10 seconds. What's the instantaneous speed at the 5th second? This problem can introduce the concept of limits in the context of everyday life.
2. **The Shrinking Chocolate Bar**: You keep breaking a chocolate bar in half. What happens to the size of the chocolate bar as you keep doing this? This can illustrate the concept of a limit approaching zero.
3. **The Infinite Series**: If you add 1/2 + 1/4 + 1/8 + 1/16 + ..., what number do you get? This problem can show how limits are used to sum infinite series.
4. **The Leaning Tower of Pisa**: If the Leaning Tower of Pisa leans one more degree each year, will it ever lay flat on the ground? This can illustrate the concept of a limit not existing.
5. **The Bouncing Ball**: A ball bounces to half its previous height each time. How high will the ball bounce after an infinite number of bounces? This can illustrate the concept of a limit approaching a finite number.
6. **The Zeno's Paradox**: You want to walk to the end of a room, but first, you must walk half the distance, then half of the remaining distance, and so on. Will you ever reach the end of the room? This ancient paradox can be solved using limits.
7. **The Speeding Rocket**: A rocket's speed increases as it gets further from Earth due to decreasing gravitational pull. What would be its speed at an infinite distance from Earth? This can illustrate the concept of a limit at infinity.
8. **The Filling Glass**: You're pouring water into a glass at a decreasing rate. Will the glass ever be completely full? This can illustrate the concept of a limit in a practical context.
9. **The Growing Tree**: A tree grows half its current height each year. What will be its height after an infinite number of years? This can illustrate the concept of a limit approaching a finite number.
10. **The Half-Life of a Radioactive Element**: A radioactive element decays by half every year. How much of the element will be left after an infinite number of years? This can illustrate the concept of a limit approaching zero.
11. **The Cooling Coffee**: A cup of coffee cools down to half the difference between its current temperature and room temperature every minute. What will be its temperature after an infinite number of minutes? This can illustrate the concept of a limit in a practical context.
12. **The Fibonacci Sequence**: The ratio of consecutive Fibonacci numbers approaches a certain number as you go further in the sequence. What is this number? This can illustrate the concept of a limit in the context of famous mathematical sequences.
13. **The Harmonic Series**: The sum of the reciprocals of all natural numbers (1 + 1/2 + 1/3 + 1/4 + ...) does not have a finite limit. This can illustrate the concept of a limit diverging.
14. **The Squeezed Lemon**: You squeeze a lemon, and each squeeze gets out half the remaining juice. How much juice will you get after an infinite number of squeezes? This can illustrate the concept of a limit in a practical context.
15. **The Infinite Ladder**: You're climbing a ladder where each rung is half the height of the previous one. Will you ever reach the top? This can illustrate the concept of a limit in a practical context.
16. **The Music Fade-Out**: A song fades out by reducing its volume by half every second. When will the song be completely silent? This can illustrate the concept of a limit approaching zero.
17. **The Population Growth**: A bacteria population doubles every hour. What will be the population after an infinite number of hours? This can illustrate the concept of a limit at infinity.
18. **The Pizza Slices**: You keep slicing a pizza into smaller and smaller pieces. What happens to the size of a piece as you keep doing this? This can illustrate the concept of a limit approaching zero.
19. **The Traveling Light**: Light travels at a constant speed. If a beam of light travels half the remaining distance to a mirror every second, when will it hit the mirror? This can illustrate the concept of a limit in a practical context.
20. **The Stock Market**: A stock's price increases by half of the remaining difference to a certain price every day. What will be the stock's price after an infinite number of days? This can illustrate the concept of a limit in a real-world financial context.
These examples and problems can make the concept of limits more relatable and interesting to high school students, helping them see the relevance of this mathematical concept in their everyday lives.
To dive deeper into one of the examples, I can ask it this:
Let's dive deeper into one of the examples: "12. **The Fibonacci Sequence**: The ratio of consecutive Fibonacci numbers approaches a certain number as you go further in the sequence. What is this number? This can illustrate the concept of a limit in the context of famous mathematical sequences." Tell me everything you know about this topic in a way that I could explain it to a high school senior and they would intuitively understand it
and it will give me this:
Sure, let's explore this fascinating aspect of the Fibonacci sequence.
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1. So, the sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on.
Now, let's look at the ratios of consecutive Fibonacci numbers:
1/1 = 1,
2/1 = 2,
3/2 = 1.5,
5/3 = 1.666...,
8/5 = 1.6,
13/8 = 1.625,
21/13 = 1.61538...
As you can see, these ratios are getting closer and closer to a certain value. This is where the concept of a limit comes in. In mathematics, a limit is the value that a function or sequence "approaches" as the input (or index) approaches some value.
In the case of the Fibonacci sequence, the ratio of consecutive numbers approaches a special number approximately equal to 1.61803, known as the golden ratio. This number is often denoted by the Greek letter phi (Φ).
The golden ratio is a mathematical constant that appears in many areas of mathematics and also in nature, art, and architecture. It's considered to be aesthetically pleasing, and shapes based on the golden ratio are often found to be more attractive.
So, when we talk about the limit of the ratio of consecutive Fibonacci numbers, we're saying that as we go further and further along the sequence, this ratio gets closer and closer to the golden ratio.
To make this concept more intuitive, you could think of it like a car driving towards a destination. The car is the ratio of consecutive Fibonacci numbers, and the destination is the golden ratio. As the car drives, it gets closer and closer to the destination, but it never quite arrives. That's what we mean when we say the limit of the ratio is the golden ratio.
I hope this explanation helps make the concept more understandable!
So, yeah, pretty cool and worthwhile to do. I don't know how this process and GPT-4 isn't extremely helpful and a value-add if used correctly.