WHWM求解数学物理题英文版开发成功

从昨天开始，我把中文版的WHWM数学物理解题思路分析大模型（简称闪问大模型，基于目前市面上已有的基础大语言模型开发）改造成英文版的大模型。下面是初步的效果。很不错。

顺便，这也启发我：任何一个基础大模型已经掌握的语言，都应该不是问题；因此，我应该考虑换一个方式来训练闪问大模型——直接用英文版的提示词和案例，然后训练好之后，让基础大模型自己来做多语言转换。

I am three years old now and my sister is twice as old as me. When she is ten years old, how many years older than me will she be?

Trans: Try to express every sentence in the problem using mathematical expressions.

Answer:

• "I am three years old now" can be expressed as:
  (where is my current age)

• "My sister is twice as old as me" means:
  (where is my sister's current age)

• "When she is ten years old, how many years older than me will she be?" indicates that when (my sister's age at time ), we need to find my age at that time and calculate the difference .

First, What: What are the objects in this problem? Specify the quantities for the objects. Which of these quantities are known, and which are to be found?

Answer:

The objects in this problem are:

1. My current age:
   (known)

2. My sister's current age:
   (related to my age, can be calculated)

3. My sister's age when she is ten years old:
   (known)

4. My age when my sister is ten years old:
   (unknown, to be found)

5. The time until my sister turns ten:
   (unknown, to be calculated)

6. The difference in age between my sister and me when she is ten:
   (unknown, to be found)

Second, How: What are the relationships between these objects? How are these relationships expressed mathematically?

Answer:

1. Calculate my sister's current age:

   years old

2. Determine the time until my sister turns ten:

   years

3. Calculate my age when my sister is ten:

   years old

4. Calculate the age difference when my sister is ten:

   years

Alternatively, we can observe that the age difference between my sister and me remains constant over time:

years

So, regardless of when, she will always be 3 years older than me.

Third, Why: Why do we focus on these objects? Why do these relationships exist between these objects? Why do we express these relationships using these mathematical expressions?

Answer:

• We focus on these objects because the problem asks for the age difference between my sister and me when she is ten years old.

• The relationships exist because:

• Age progression: Both my sister and I age at the same rate; every year, each of us becomes one year older. So, our ages increase linearly over time.

• Current age relation: My sister's current age is twice mine, which is a key piece of information to calculate her current age.

• We use these mathematical expressions because:

• Addition () represents the passage of time and accumulation of years in age.

• Subtraction () calculates the difference in our ages.

• Multiplication () reflects the "twice as old" relationship between my sister and me.

• By expressing these relationships mathematically, we can perform precise calculations to find the unknown quantities.

Fourth, Meaningful: What have I gained from solving this problem? Divide into four sub-questions and sub-steps.

• First, Validation: Check whether the problem has been solved.

• We found that when my sister is ten years old (), I will be seven years old ().

• The age difference at that time is years.

• Currently, the age difference is years.

• Since the age difference remains consistent, our calculations are correct, and the problem is solved.

• Validation:

• Second, Logic Summary: Summarize the logic of the solution method.

• Logic Summary:

1. Calculated my sister's current age using the given relationship that she is twice as old as me.

2. Determined the time it takes for my sister to reach ten years old.

3. Calculated my age at the time when my sister is ten years old.

4. Computed the age difference at that future time.

5. Noted that the age difference remains constant over time.

• The logic is correct; each step logically follows from the previous one, and all calculations are consistent.

• An alternative method is to recognize immediately that the age difference between siblings remains constant. Therefore, my sister will always be 3 years older than me.

• Reflection:

• Understanding of Age Differences: I have reinforced the concept that the age difference between two people remains the same over time if both age at the same rate.

• Mathematical Modeling: I learned how to translate word problems into mathematical expressions systematically.

• Problem-Solving Strategy: Applying the WHWM method helps in breaking down the problem into manageable parts and ensures a thorough understanding.

• Efficiency in Problem Solving: Recognizing inherent properties (like constant age difference) can simplify calculations and make problem-solving more efficient.

• Lesson Learned:

By applying the WHWM analysis, I systematically solved the problem, ensuring that each step was logically sound and that I thoroughly understood the relationships between the different elements of the problem.

There are two black bags containing glass marbles, with a total of 300 marbles. I cannot open the bags to count them, but I know that the larger bag holds twice as many marbles as the smaller bag. How many marbles are in each bag?

Trans: Try to express every sentence in the problem using mathematical expressions.

Answer:

• "There are two black bags containing glass marbles, with a total of 300 marbles" means:

  where is the number of marbles in the larger bag, and is the number of marbles in the smaller bag.

• "The larger bag holds twice as many marbles as the smaller bag" means:

• "How many marbles are in each bag?" means we need to find the values of and .

First, What: What are the objects in this problem? Specify the quantities for the objects. Which of these quantities are known, and which are to be found?

Answer:

The objects in this problem are:

• Total number of marbles ():

• Known quantity.
• marbles.

• Number of marbles in the larger bag ():

• Unknown quantity.
• To be found.

• Number of marbles in the smaller bag ():

• Unknown quantity.
• To be found.

Second, How: What are the relationships between these objects? How are these relationships expressed mathematically?

Answer:

We have the following relationships:

1. Total marbles equation:

2. Relation between the larger and smaller bag:

Solving the equations:

• Substitute from the second equation into the first equation:

• Simplify the equation:

• Solve for :

• Find :

Third, Why: Why do we focus on these objects? Why do these relationships exist between these objects? Why do we express these relationships using these mathematical expressions?

Answer:

• Why we focus on these objects:

• The problem asks for the number of marbles in each bag, so we need to determine and .

• Why these relationships exist:

• The total marbles equation () exists because the total number of marbles is the sum of marbles in both bags. This uses the concept of addition (combining quantities).
• The relation between the bags () exists because the larger bag contains twice as many marbles as the smaller bag. This uses the concept of multiplication (repeated addition or scaling).

• Why we use these mathematical expressions:

• They allow us to model the problem precisely and perform calculations to find the unknown quantities.
• Mathematical expressions provide a clear and concise way to represent relationships and solve for variables using algebraic methods.

Fourth, Meaningful: From this problem's solution, what have I gained? Divide into four sub-questions and sub-steps.

1. First, Validation: Check whether the problem has been solved.

• Check the number of marbles in each bag:

• Verify the total marbles:

• Verify the larger bag has twice as many marbles:

• Both conditions are satisfied, so the problem has been correctly solved.

• Smaller bag:
• Larger bag:
• This matches the total of 300 marbles.
• This matches the calculated value for .

• Validation:

• Translation: Converted the word problem into mathematical equations representing the relationships between quantities.
• Identification: Recognized the known and unknown variables in the problem.
• Substitution: Used substitution to reduce the system of equations to one variable.
• Calculation: Solved for and then calculated .
• Verification: Validated the solution by checking that it meets all the given conditions.

• Logic Summary:

• The logic is correct; each step follows logically and mathematically.
• Alternative Method:
• Consider the smaller bag as 1 unit and the larger bag as 2 units (since it has twice as many marbles).
• Total units:
• Calculate marbles per unit:
• Calculate marbles in each bag:
• This method reaches the same conclusion and emphasizes understanding ratios.
• Smaller bag:
• Larger bag:
• Unit Method:

• Reflection:

• Translating Word Problems: Improved ability to convert verbal descriptions into mathematical equations.
• Algebraic Techniques: Reinforced understanding of solving systems of equations using substitution.
• Concept of Ratios and Proportions: Gained a deeper appreciation for how ratios can simplify problem-solving.
• Problem-Solving Strategy: Recognized the importance of identifying known and unknown variables and systematically applying relationships.
• Application of WHWM Method: Found that the WHWM framework aids in thoroughly analyzing and solving problems by breaking them down into manageable steps.

• Lesson Learned:

By applying the WHWM analysis, I have successfully solved the problem and enhanced my understanding of mathematical problem-solving techniques.

Two blocks, a wooden block and a metal block, with a volume ratio of 8:3, are tied together with a rope and placed in water. It is observed that of the wooden block's volume is above the water surface. After cutting the rope, half of the wooden block's volume is above the water surface. What is the density of the metal block?

Trans: Read the problem and translate it into academic language, converting each sentence into physical concepts, laws, and mathematical expressions.

Answer:
"Two blocks, a wooden block and a metal block, with a volume ratio of 8:3, are tied together with a rope and placed in water." This means that the volume of the wooden block () and the metal block () satisfy the ratio , or specifically, and , where is a unit volume.

"It is observed that of the wooden block's volume is above the water surface." This implies that the submerged volume of the wooden block () is .

"After cutting the rope, half of the wooden block's volume is above the water surface." This means that the submerged volume of the wooden block becomes after the rope is cut.

"What is the density of the metal block?" indicates that the density of the metal block () is the required unknown to be determined.

What: Identify the physical objects involved in the problem, specify their physical quantities, their values, and determine which are known and which need to be found.

Answer:
The physical objects and their associated quantities are:

• Wooden Block

• Volume: (known)
• Density: (unknown)
• Submerged Volume (initial): (known)
• Submerged Volume (after cutting the rope): (known)
• Weight: (unknown)

• Metal Block

• Volume: (known)
• Density: (unknown, required)
• Weight: (unknown)

• Water

• Density: (known)
• Acceleration due to gravity: (known)

• Rope

• Tension: (unknown, relevant before cutting the rope)

• System as a Whole (before cutting the rope)

• Total Weight: (unknown)
• Total Submerged Volume: (known)

How: Describe the relationships between the objects, the physical laws governing these relationships, and express these laws using mathematical equations. Perform calculations starting from known quantities towards the unknown.

Answer:
This scenario involves the principles of buoyancy, specifically Archimedes' Principle, and the equilibrium of forces.

1. Before Cutting the Rope:

• Buoyant Force Balance:

  Substituting the known submerged volume:

  Simplifying by canceling :

  Substituting and :

  Dividing both sides by :

• Buoyant Force Balance for the Wooden Block:

  Substituting the known submerged volume:

  Simplifying by canceling and substituting :

  Dividing both sides by :

  Simplifying:

• Substitute and into Equation 1:

Therefore, the density of the metal block is approximately .

Why: Explain why these objects are of interest, why they are related in this way, why they obey these physical laws, and why these relationships are expressed mathematically in this manner.

Answer:
This problem involves understanding buoyancy and the equilibrium of forces acting on submerged objects. The objects of interest—the wooden block and the metal block—are connected via the rope and placed in water, making the study of their buoyant interactions essential for determining the density of the metal block.

• Buoyant Force and Archimedes' Principle:
  Archimedes' Principle states that the buoyant force on a submerged object is equal to the weight of the displaced fluid. This principle is fundamental in analyzing how objects float or sink in a fluid based on their densities relative to the fluid.

• Force Equilibrium:
  The condition of equilibrium—where the buoyant force equals the total weight of the objects—allows us to set up equations that relate the densities and volumes of the blocks. When the rope is intact, the combined system of the wooden and metal blocks must displace an amount of water equal to their total weight. After cutting the rope, only the wooden block's buoyancy is directly considered, altering the displacement and thus providing another equation to solve for the unknown density.

• Mathematical Representation:
  Translating physical laws into mathematical equations enables precise calculations and quantification of the relationships between different physical quantities. By expressing volumes, densities, and forces mathematically, we can apply algebraic methods to solve for unknowns systematically.

• Dependence on Material Properties:
  The differing behaviors of the blocks before and after cutting the rope—whether parts of the wooden block are submerged or not—highlight the importance of material density in buoyancy. Understanding these relationships is crucial for applications in engineering, shipbuilding, and material science.

Thus, the relationships between volume ratios, density, and buoyant force are governed by fundamental physical laws, allowing us to determine the unknown density of the metal block through systematic analysis.

Meaningful: From solving this problem, what have I gained? Break it down into four sub-steps.

• Validation: Check that the problem has been solved correctly.

  Answer:
  The calculated density of the metal block is approximately . To validate:

• Before Cutting the Rope:

  Substituting the known values:

• After Cutting the Rope:

• Overall Validation:
  Both scenarios confirm that the calculations are consistent with the given conditions and physical laws.

• Logic Summary: Organize and summarize the logical flow of the solution.

  Answer:

  This logical progression from understanding volume ratios to applying buoyancy principles and solving simultaneous equations ensured a coherent and structured approach to finding the unknown density.

1. Establish Volume Ratios: Recognized the volume ratio between the wooden and metal blocks as 8:3.

2. Analyze Buoyant Forces Before Cutting the Rope: Applied Archimedes' Principle to relate the submerged volumes and total weight, forming the first equation.

3. Determine Wood Density After Cutting the Rope: Utilized the new submerged volume to derive the density of the wooden block.

4. Solve for Metal Density: Substituted the known wood density and water density into the first equation to solve for the metal block's density.