A container can fit exactly 5 basketballs, 25 softballs, or 125 table tennis balls. 2 basketballs, 10 softballs and some table tennis balls are put into such a container. What is the greatest possible number of table tennis balls that can fit into the container?
Translations
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Credits
weelookang@gmail.com; Francisco Esquembre; Felix J. Garcia Clemente
Main Theme: Both sources revolve around a specific mathematical word problem concerning the equivalent volume occupancy of different types of balls within a container. The core question asks to determine the maximum number of table tennis balls that can fit in a container already holding a certain number of basketballs and softballs, given the container's capacity in terms of each ball type.
Key Ideas and Facts:
1. The Word Problem:
- The central problem states a container's capacity in terms of three different objects: basketballs, softballs, and table tennis balls. The equivalencies provided are:
- 5 basketballs = Full Container
- 25 softballs = Full Container
- 125 table tennis balls = Full Container
- The problem then introduces a scenario where some basketballs and softballs are already in the container:
- 2 basketballs
- 10 softballs
- The question to be answered is: "What is the greatest possible number of table tennis balls that can fit into the container?"
2. Implicit Concept of Volume Equivalence:
- The problem implicitly establishes a concept of volume equivalence. The fact that the container can hold a specific number of each type of ball suggests that these quantities occupy the same total volume. This allows for the conversion between the "volume units" of each ball type.
3. Visualization as a Teaching Tool:
- The second source explicitly highlights the use of visualization as a method to understand and solve this problem. It describes an HTML5 JavaScript tool designed for this purpose.
- The tool allows users to "click on the first part of the question for the visualization to appear. You should see 5 basketball, 25 softball and 125 table tennis balls." This visual representation aims to make the volume equivalence concrete.
- For the second part of the problem, the tool provides a "visualization hint which can be shown on click, you should see 2 basketball and 10 softballs arranged from bottom to top, leaving the top RED color as the maximum number of table tennis balls you can add to the container." The red area visually represents the remaining capacity.
4. Interactive Features of the Tool:
- The visualization tool offers interactive features to enhance learning:
- "Students can drag to recentre and mouse scroll to zoom in, phone will be pinch in and out." This allows for better examination of the visual representation.
- There is a "check answer input field for testing your own understanding." This promotes self-assessment.
5. Adaptability for Concept Building:
- The tool is designed to be flexible for educational purposes: "The variables can be changed to a different context for stronger concept building." This suggests that teachers can modify the numbers and types of objects to create similar problems and reinforce the underlying mathematical concepts.
6. Credits and Development:
- Both sources credit weelookang@gmail.com; Francisco Esquembre; Felix J. Garcia Clemente for their work in creating the problem and the visualization tool.
- The tool was "Compiled with EJS 6.0 (191124)" (Easy Java/JavaScript Simulations Toolkit), which is further credited to Francisco Esquembre and Félix Jesús Garcia Clemente.
- Open Educational Resources / Open Source Physics @ Singapore is identified as the platform hosting and distributing the visualization tool.
7. Pedagogical Approach:
- The sources indicate a pedagogical approach that emphasizes visual learning and interactive engagement to help students understand mathematical concepts, specifically those involving ratios and proportions applied to volume/capacity. The "For Teachers" section explicitly guides educators on how to use the visualization tool effectively.
Quotes:
- Problem Statement: "A container can fit exactly 5 basketballs, 25 softballs, or 125 table tennis balls. 2 basketballs, 10 softballs and some table tennis balls are put into such a container. What is the greatest possible number of table tennis balls that can fit into the container?" (Source 1 & 2)
- Visualization for Equivalence: "In this question, you can click on the first part of the question for the visualization to appear. You should see 5 basketball, 25 softball and 125 table tennis balls. The equivalent is shown via the same height of the drawing. This helps to establish the first part of the question." (Source 2)
- Visualization for Remaining Capacity: "The second part has the visualization hint which can be shown on click, you should see 2 basketball and 10 softballs arranged from bottom to top, leaving the top RED color as the maximum number of table tennis balls you can add to the container." (Source 2)
- Interactive Features: "Students can drag to recentre and mouse scroll to zoom in, phone will be pinch in and out." (Source 2)
- Adaptability: "The variables can be changed to a different context for stronger concept building." (Source 2)
Conclusion:
The provided sources focus on a specific word problem involving the volume equivalence of different balls within a container. They highlight not only the mathematical challenge of solving the problem but also the development and application of an interactive visualization tool designed to aid understanding. The tool leverages visual representation and interactive features to make the abstract concept of volume equivalence more tangible for students, while also offering flexibility for educators to adapt the problem for various learning contexts. The credits indicate a collaborative effort in developing both the problem and the educational resource.
Study Guide: Container Capacity Problem
Core Concept: Understanding proportional relationships and unit equivalence to solve a capacity problem.
Key Information from Sources:
- A container has a fixed total capacity.
- This capacity can be filled by exactly 5 basketballs, OR 25 softballs, OR 125 table tennis balls. This establishes the equivalence between the volumes of these different balls within the container.
- A specific number of basketballs (2) and softballs (10) are already placed in the container.
- The problem asks for the greatest possible number of table tennis balls that can fit in the remaining space.
- Visualizations are available to demonstrate the equivalence and the remaining space.
Study Points:
- Establish Equivalence: Understand the ratio of volumes between the different types of balls. If 5 basketballs = 1 container, and 125 table tennis balls = 1 container, then 5 basketballs are equivalent in volume to 125 table tennis balls. You should be able to determine the table tennis ball equivalent for one basketball and one softball.
- Calculate Occupied Space: Determine what fraction or proportion of the container's capacity is already filled by the 2 basketballs and 10 softballs. To do this effectively, you'll need to express the volume of the basketballs and softballs in terms of a common unit, such as the equivalent number of table tennis balls.
- Determine Remaining Capacity: Subtract the occupied capacity (in table tennis ball equivalents) from the total capacity of the container (in table tennis ball equivalents).
- Final Answer: The remaining capacity, expressed in the number of table tennis balls, is the greatest possible number of table tennis balls that can still fit in the container.
- Visualization as a Tool: Recognize how the provided visualizations (showing the balls fitting in the container and the layering of basketballs and softballs) can aid in understanding the problem and verifying your calculations.
Quiz: Short Answer Questions
- According to the problem statement, what is the total capacity of the container in terms of the number of softballs it can hold?
- If 5 basketballs have the same volume as 125 table tennis balls, how many table tennis balls have the same volume as one basketball? Show your reasoning.
- The problem mentions a visualization where 2 basketballs and 10 softballs are arranged. What does the red color at the top of this visualization represent?
- What common unit is most helpful to use when calculating the total occupied volume of the container by the basketballs and softballs? Why?
- If one softball has the same volume as 5 table tennis balls, what is the equivalent volume of 10 softballs in terms of table tennis balls?
- Two basketballs are placed in the container. Using your answer from question 2, what is the equivalent volume of these two basketballs in terms of table tennis balls?
- What fraction of the container's capacity is occupied by one basketball? Express this as a fraction where the denominator represents the equivalent number of table tennis balls the container can hold.
- After placing 2 basketballs and 10 softballs in the container, how much of the container's capacity (in terms of the number of table tennis balls) is still available? Show your calculation.
- The problem asks for the "greatest possible number" of table tennis balls. Why is this phrasing used, assuming the balls cannot be compressed?
- How can the interactive features mentioned in the source, such as dragging to recenter and mouse scrolling to zoom, help a student understand this type of problem?
Quiz Answer Key
- The container can fit exactly 25 softballs. This information is directly stated in the problem.
- One basketball has the same volume as 25 table tennis balls. This is because 5 basketballs = 125 table tennis balls, so dividing both sides by 5 gives 1 basketball = 25 table tennis balls.
- The red color at the top of the visualization represents the remaining volume or capacity of the container after the 2 basketballs and 10 softballs have been placed inside.
- Table tennis balls are the most helpful common unit because the final answer is requested in terms of table tennis balls, and the total capacity is already given in terms of this unit (125).
- 10 softballs have the same volume as 50 table tennis balls. Since 1 softball = 5 table tennis balls (25 softballs = 125 table tennis balls), then 10 softballs = 10 * 5 = 50 table tennis balls.
- Two basketballs have the same volume as 50 table tennis balls. Since 1 basketball = 25 table tennis balls, then 2 basketballs = 2 * 25 = 50 table tennis balls.
- One basketball occupies 1/5 of the container's capacity in terms of basketballs, which is equivalent to 25/125 of the container's capacity in terms of table tennis balls (or 1/5).
- The remaining capacity is 25 table tennis balls. The 2 basketballs occupy the equivalent of 50 table tennis balls, and the 10 softballs occupy the equivalent of 50 table tennis balls, totaling 100 table tennis ball equivalents. Therefore, 125 (total capacity) - 100 (occupied) = 25 table tennis balls.
- The phrasing "greatest possible number" is used because we assume the table tennis balls are placed to maximize the space used and cannot be compressed or distorted to fit more than their equivalent volume allows.
- The interactive features allow students to get a better visual and spatial understanding of how the different sized balls fit within the container, aiding in their intuitive grasp of the volume equivalences and the concept of remaining space.
Essay Format Questions
- Explain the concept of volume equivalence as it applies to the basketball, softball, and table tennis ball problem. Discuss how establishing these equivalences is crucial for solving the problem.
- Describe the role of visualization tools in understanding and solving mathematical word problems like the container capacity problem. How does the specific visualization mentioned in the sources aid in finding the solution?
- Discuss how this type of problem can be adapted to teach concepts of fractions, ratios, and proportions. Provide specific examples of how the numbers or objects could be changed to illustrate these concepts.
- Analyze the steps involved in solving the container capacity problem. Explain the logical reasoning behind each step and why it is necessary to arrive at the correct answer.
- Consider the educational value of using real-world scenarios, like filling a container with different objects, to teach mathematical concepts. What are the benefits and potential challenges of this approach?
Glossary of Key Terms
- Capacity: The maximum amount that a container can hold. In this problem, it is a fixed volume.
- Equivalence: Having the same value, measure, or function. In this context, it refers to different quantities of balls occupying the same volume.
- Ratio: A comparison of two quantities by division. The relationship between the number of each type of ball that fits in the container can be expressed as a ratio.
- Proportion: A statement that two ratios are equal. The equivalence between the balls establishes proportions in terms of volume.
- Visualization: The representation of data or concepts in a graphical or pictorial form to aid understanding. In this case, a visual representation of the balls in the container.
- Unit Equivalence: The idea that one unit of a certain item can be equal to a specific number of units of another item (e.g., 1 basketball is equivalent to 25 table tennis balls in volume).
- Occupied Volume: The amount of space within the container that is currently filled by the objects placed inside.
- Remaining Capacity: The amount of space left within the container after some objects have been placed inside.
- Greatest Possible Number: The maximum quantity of an item that can fit into the remaining space under given constraints (in this case, assuming the balls cannot be compressed).
- Open Educational Resources (OER): Teaching and learning materials that are freely available online for anyone to use, adapt, and share.
For Teachers
In this question, you can click on the first part of the question for the visualization to appear. You should see 5 basketball, 25 softball and 125 table tennis balls.
The equivalent is shown via the same height of the drawing. This helps to establish the first part of the question.
The second part has the visualization hint which can be shown on click, you should see 2 basketball and 10 softballs arranged from bottom to top, leaving the top RED color as the maximum number of table tennis balls you can add to the container.
Students can drag to recentre and mouse scroll to zoom in, phone will be pinch in and out.
Finally, there is a check answer input field for testing your own understanding.
The variables can be changed to a different context for stronger concept building.
Enjoy!
Research
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Frequently Asked Questions: Container Capacity Problem
1. What is the central problem being addressed in the provided sources?
The core problem revolves around determining the remaining capacity of a container after it has been partially filled with items of different sizes. Specifically, the question asks for the maximum number of table tennis balls that can fit into a container already holding a certain number of basketballs and softballs.
2. What information is crucial for solving this type of problem?
To solve this problem, it's essential to know the relative volumes or space occupied by each type of item. The sources explicitly state the container can fit exactly 5 basketballs, 25 softballs, or 125 table tennis balls. This establishes the equivalence in terms of the container's total capacity.
3. How can we establish the relationship between the different types of balls in terms of container capacity?
The problem implies that the space occupied by 5 basketballs is equal to the space occupied by 25 softballs, which is also equal to the space occupied by 125 table tennis balls. This allows us to find conversion factors between the number of each type of ball that fits in the container.
4. How much of the container's capacity is used by one basketball?
Since 5 basketballs fill the container, one basketball occupies 1/5 of the container's total capacity.
5. How much of the container's capacity is used by one softball?
Since 25 softballs fill the container, one softball occupies 1/25 of the container's total capacity.
6. How much of the container's capacity is used by the initially placed basketballs and softballs?
The problem states that 2 basketballs and 10 softballs are placed in the container. Two basketballs occupy 2 * (1/5) = 2/5 of the capacity. Ten softballs occupy 10 * (1/25) = 10/25 = 2/5 of the capacity. Therefore, the basketballs and softballs together occupy 2/5 + 2/5 = 4/5 of the container's total capacity.
7. How can we determine the remaining capacity for table tennis balls?
If 4/5 of the container is already filled, the remaining capacity is 1 - 4/5 = 1/5 of the container's total capacity.
8. Given the remaining capacity, how many table tennis balls can fit?
We know that the entire container can hold 125 table tennis balls, which represents 1 (or 5/5) of the capacity. Since the remaining capacity is 1/5 of the total, we can fit (1/5) * 125 = 25 table tennis balls into the remaining space. The greatest possible number of table tennis balls that can fit is 25.
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- Category: 3. Multiplication and Division
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