Mathematics and AI Chatbots
Artificial Intelligence is no longer the talk of the future, it’s already here and despite all the negativity the Hollywood movies have spread about the robots and artificial intelligence it’s taking the center stage in every walk of life, whether it’s industry, community or education it’s being embraced by everyone. Despite being skeptical about it’s impact and its consequences in the near and far future, it’s being accepted as another perspective to the daily life routines and chores.
ChatGPT, Bard/Gemini, Gork, Claude, Copilot, Llama, ChatSonic these are the big names in conversational ai tools based on the following characteristics: these are customizable, scalable, and compatible with multiple platforms. Besides these they are supposed to be accurate, be highly responsive, and be able to interact like humans. When you are in a conversation with an AI chatbot you should get the feeling of having a conversation with a coworker not a machine. OpenAI claims that chatgpt is able to handle problems related to mathematics and algebra, so does Google claims about Gemini.
Being a math and Algebra teacher, I wanted to check how Chat GPT 3.5/4.0 and BARD/Gemini solve questions related to algebraic concepts. My intent was to explore accuracy, explanations, and Graphical representations. I chose three basic problems that we usually come across in Algebra.
- Solving one variable linear equations (Content and Procedure)
- Solving Simultaneous equations (Problem Solving and Reasoning)
- Solving an exponential Decay Problem (Problem Solving)
PROBLEM 1: SOLVING ONE VARIABLE LINEAR EQUATION
Solve the algebraic equation 4x + 3(x — 6) + 24 = 2(3x — 2) + 2x -15
ChatGPT 4.0 RESPONSE
GEMINI RESPONSE
There is a discrepancy in the answer given by Gemini, whereas ChatGPT 4.0 and 3.5 both gave the correct answer.
PROBLEM 2: SOLVING SIMULTANEOUS EQUATIONS
Ahmed and Ali are on a road trip. Each of them drives at a constant speed. Ahmed is a safe driver and travels 50 miles per hour for the entire trip. Ali is not a safe driver. He drives 75 miles per hour throughout the trip. Ahmed and Ali left from the same location, but Ahmed left at 7:00 a.m., and Ali left at 10:00 a.m. Assuming they take the same route, will Ali ever catch up to Ahmed? If so, approximately when?
a. Write the linear equation that represents Ahmed’s constant speed. Make sure to include in your equation the extra time that Ahmed was able to travel.
b. Write the linear equation that represents Ali’s constant speed.
c. Write the system of linear equations that represents this situation.
d. Sketch the graphs of the two linear equations.
RESPONSE FROM CHATGPT 4.0
Let’s address each part of this problem step by step:
a. Ahmed’s Linear Equation
Ahmed’s speed is constant at 50 miles per hour. If we let \( t \) represent the time in hours after 7:00 a.m., then the distance \( d \) that Ahmed travels can be expressed as:
d = 50t
Since Ahmed starts 3 hours before Ali, we should account for this head start in his travel equation. Since he travels for 3 hours before Ali starts, he covers 50 X 3 = 150 miles. Therefore, the equation accounting for his head start is:
d = 50t + 150
b. Ali’s Linear Equation
Ali’s speed is constant at 75 miles per hour. Unlike Ahmed, Ali starts at 10:00 a.m., so his time t in the equation starts from zero at 10:00 a.m. The distance d that Ali travels can be expressed as:
d = 75t
Notice, there’s no initial distance added to Ali’s equation because he starts from the same location but later in time.
c. System of Linear Equations
The situation can be represented by the following system of linear equations where d represents the distance each has traveled from the starting point:
1. Ahmed’s distance: d = 50t + 150
2. Ali’s distance: d = 75t
d. Sketching the Graphs
To sketch the graphs, we will represent the time after 7:00 a.m. on the x-axis and the distance traveled on the y-axis. Ahmed’s graph will start at (0,150) because of his 3-hour head start and have a slope of 50. Ali’s graph will start at (0,0) and have a slope of 75.
Let’s now plot these equations to visualize the scenario and find out if and when Ali catches up to Ahmed.
The graph above illustrates the distances traveled by Ahmed and Ali over time after 7:00 a.m. Ahmed’s distance starts at 150 miles due to his 3-hour head start and increases linearly. Ali’s distance starts from zero and also increases linearly but at a steeper rate due to his higher speed.
From the graph, we can see that Ali’s line (red) intersects Ahmed’s line (blue), indicating that Ali does catch up to Ahmed at a certain point in time after 7:00 a.m. The exact time at which Ali catches up can be calculated by setting the two distance equations equal to each other and solving for t, but visually, it’s clear from the graph that the catch-up occurs at (6, 450).
GEMINI’s RESPONSE
Ahmed vs. Ali: The Road Trip Chase
a. Ahmed’s Equation:
Let d be the distance traveled and t be the time in hours. Ahmed travels at a constant speed of 50 mph and starts at 7:00 AM. Since 10:00 AM is 3 hours later, Ahmed has a head start of 3 hours * 50 mph = 150 miles. Therefore, his equation is:
d = 50t + 150
b. Ali’s Equation:
Ali travels at a constant speed of 75 mph but starts 3 hours later than Ahmed (at 10:00 AM). So, his equation is:
d = 75(t — 3) (subtracting 3 from t to account for the later start)
c. System of Equations:
The system of equations representing this situation is:
d = 50t + 150 d = 75(t — 3)
d. Sketching the Graphs:
- Ahmed’s graph: A straight line with a slope of 50 and a y-intercept of 150. It starts at (0, 150) and goes upwards.
- Ali’s graph: Another straight line with a steeper slope of 75. However, it starts at (3, 225) due to the later departure (3 hours * 75 mph). It also goes upwards.
Note: In this scenario, the graphs will not intersect, meaning Ali will never catch up to Ahmed. This is because Ahmed’s head start is too large, and even though Ali travels faster, he cannot overcome the initial distance gap.
RESPONSE FROM CHATGPT 3.5
We can clearly say the chatgpt 4.0 has given a detailed answer with explanations accurately. Whereas Gemini has not dealt with the problem correctly.
PROBLEM 3: EXPONENTIAL DECAY QUESTION
A construction company purchased some equipment costing $500000. The value of the equipment depreciates (decreases) at a rate of 15% per year. a. Write a formula that models the value of the equipment each year. b. What is the value of the equipment after 12 years? c. Graph the points (t, v(t)) for integer values of 0 ≤ t ≤ 20.
ChatGPT 4.0 Response
GEMINI RESPONSE
CHATGPT 3.5 RESPONSE
Both have accurately solved the question and given a step by step procedure. Gemini cannot create graphs so it has given two methods to create the graphs. Chatgpt 3.5 cannot create graphs and
ANALYSIS
This bring us to the fact that these tools are not for the novice people. If a students uses it for homework, without verifying and checking the answers then it’s leading the student to misconceptions and wrong submissions. If a teacher is using it for generating questions with solutions then they have to verify the solutions because teacher is the authority in his/her craft. These tools are excellent for generating questions for assessments, as educators are always on the lookout for new material for assessments and are striving to improve their assessments.
These AI Tools can handle questions related to mathematics and algebra to some extent, because these tools were basically trained on LLMs that can handle text better. The good news is, we can train them on solving mathematics problems, if we keep on giving challenging math problems and correcting the response given by the chatbots. To get the desired results you have to prompt better and it's an iterative process and we need to go back and forth with our prompts to get the desired results. This is the way to train these chatbots to solve the unsolved problems and mysteries of the world of science.
This is an interesting article that I came across when I wanted to dive deeper into mathematics and AI: Chatbots Put to the Test in Math and Logic Problems
