Fraction Division & Inequality City Design
Created bySheri Collier
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Fraction Division & Inequality City Design

Grade 6Math1 days
★★★★★
4.0 (1 rating)
In this project, 6th-grade students design a scaled model city, applying their knowledge of fraction division, algebraic expressions, and inequalities to meet real-world constraints. Students compute quotients of fractions, use variables to represent numbers, and write/interpret inequalities. They create a detailed construction plan, build the model, and document their design choices, mathematical calculations, and encountered challenges in a final report. This project integrates mathematical concepts with real-world applications, fostering a deeper understanding of scaling and problem-solving.
Fraction DivisionAlgebraic ExpressionsInequalitiesScaled ModelReal-World ApplicationCity Design
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Inquiry Framework

Question Framework

Driving Question

The overarching question that guides the entire project.How can we design a scaled model incorporating fractional relationships, algebraic expressions, and inequalities to meet specific real-world constraints and represent the solutions visually?

Essential Questions

Supporting questions that break down major concepts.
  • How can fractions be used to solve real-world problems?
  • How do variables help us represent real-world situations with mathematical expressions and inequalities?
  • How can inequalities be used to describe limitations or conditions in real-world scenarios?
  • How can number lines visually represent solutions to inequalities?

Standards & Learning Goals

Learning Goals

By the end of this project, students will be able to:
  • Students will be able to compute quotients of fractions and solve related word problems.
  • Students will be able to use variables to represent numbers and write algebraic expressions.
  • Students will be able to write and interpret inequalities to represent constraints.
  • Students will be able to visually represent solutions to inequalities on a number line.
  • Students will be able to design a scaled model incorporating fractional relationships.
  • Students will be able to apply mathematical concepts to real-world scenarios through project design.

Common Core Standards

6.NS.A.1
Primary
Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.Reason: Directly involves fractional relationships and problem-solving.
6.EE.B.6
Primary
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.Reason: Focuses on using variables to represent numbers and write expressions.
6.EE.B.8
Primary
Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagramsReason: Involves writing and interpreting inequalities to represent constraints.

Entry Events

Events that will be used to introduce the project to students

The Great Candy Conundrum

A local candy store needs help optimizing its packaging for shipping. They present students with oddly shaped candies and shipping restrictions, challenging them to design the most efficient box using fraction division and inequalities. This connects math to a real-world business problem and sparks immediate interest.
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Portfolio Activities

Portfolio Activities

These activities progressively build towards your learning goals, with each submission contributing to the student's final portfolio.
Activity 1

Fraction Frenzy: Dividing to Conquer Dimensions

Students begin by exploring fraction division in the context of scaling dimensions. They'll calculate how many times smaller a scaled-down object is compared to its original size, focusing on fractional relationships.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Choose three real-world objects (e.g., a book, a table, a car). Measure their length, width, and height (or diameter, if applicable) in inches or centimeters.
2. Decide on a scale factor that is a fraction (e.g., 1/2, 1/4, 1/3). This will be the factor by which you reduce each dimension.
3. Divide each original dimension by the denominator of your chosen scale factor. For example, if your scale factor is 1/4, divide each dimension by 4.
4. Record the new, scaled-down dimensions for each object. Create a table showing the original and scaled dimensions.
5. Write a paragraph explaining how fraction division was used to determine the scaled dimensions and what this means in terms of the object's new size compared to the original.

Final Product

What students will submit as the final product of the activityA table comparing original and scaled dimensions of three objects, along with a paragraph explaining the process and results.

Alignment

How this activity aligns with the learning objectives & standardsAligns with 6.NS.A.1, focusing on computing quotients of fractions in a real-world context.
Activity 2

Variable Voyage: Charting Unknown Territories

Students transition to using variables to represent the scaled dimensions. They'll create algebraic expressions to represent the scaling process, reinforcing the concept of a variable as an unknown number.

Steps

Here is some basic scaffolding to help students complete the activity.
1. For one of the objects from the previous activity, assign variables to represent the original dimensions (e.g., l = length, w = width, h = height).
2. Using the same scale factor as before (e.g., 1/4), write algebraic expressions to represent the scaled dimensions. For example, if the original length is 'l' and the scale factor is 1/4, the scaled length would be 'l/4'.
3. Choose a different scale factor (again, a fraction) and repeat step 2, creating new expressions for the scaled dimensions with the new scale factor.
4. Substitute the actual values of the original dimensions (from Activity 1) into your algebraic expressions to calculate the numerical values of the scaled dimensions. Verify that these match your previous calculations.
5. Write a reflection on how variables and algebraic expressions can efficiently represent the scaling process, regardless of the specific dimensions or scale factor.

Final Product

What students will submit as the final product of the activityAlgebraic expressions representing scaled dimensions with different scale factors, calculations showing the expressions match previous results, and a reflection on the use of variables.

Alignment

How this activity aligns with the learning objectives & standardsAligns with 6.EE.B.6, focusing on using variables to represent numbers and write expressions when solving a real-world problem.
Activity 3

Inequality Island: Setting the Boundaries

Students explore inequalities by setting constraints on the scaled model. They'll write and interpret inequalities to represent limitations on size or cost, and represent the solutions on a number line.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Imagine you have a constraint on the maximum length of one dimension of your scaled object (e.g., the scaled length must be less than 5 inches). Write an inequality to represent this constraint, using the variable you defined in Activity 2.
2. Solve the inequality to find the possible range of values for the original dimension that would satisfy the constraint on the scaled dimension.
3. Draw a number line and represent the solution set of the inequality on the number line. Use an open or closed circle appropriately to indicate whether the endpoint is included in the solution.
4. Introduce a second constraint, such as a minimum size requirement (e.g., the scaled width must be greater than 2 inches). Write another inequality to represent this constraint and solve it.
5. Represent the solution set of the second inequality on the same number line (or a new one). Identify the range of values that satisfy both inequalities simultaneously. Explain what this means in the context of your scaled object.

Final Product

What students will submit as the final product of the activityInequalities representing constraints on scaled dimensions, solutions to the inequalities, number line representations of the solutions, and an explanation of how the inequalities limit the possible sizes of the original object.

Alignment

How this activity aligns with the learning objectives & standardsAligns with 6.EE.B.8, focusing on writing and interpreting inequalities to represent constraints and representing solutions on number line diagrams.
Activity 4

Model Masterpiece: Scaling to Perfection

Students combine their knowledge of fraction division, algebraic expressions, and inequalities to design a scaled model that meets specific criteria and constraints. This is the culmination of the project, where they apply their learning to a tangible product.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Choose an object to model. This could be a building, a vehicle, a piece of furniture – anything that can be scaled down.
2. Determine the actual dimensions of the object. Research or measure as needed.
3. Establish a scale factor for your model. This could be a fraction (e.g., 1/12 for a scale model of a building) or a ratio (e.g., 1 inch = 1 foot). Justify why you chose this scale.
4. Calculate the dimensions of your scaled model using fraction division (as in Activity 1) and algebraic expressions (as in Activity 2).
5. Identify at least two constraints on your model, such as a maximum height, a minimum width, or a cost limitation. Write inequalities (as in Activity 3) to represent these constraints.
6. Create a detailed plan for building your scaled model, including materials, tools, and construction steps. The plan should demonstrate how your model meets the chosen constraints.
7. Build your scaled model. Document the process with photos or videos.
8. Write a final report explaining your design choices, the mathematical calculations you used, how your model meets the constraints, and any challenges you encountered during the construction process.

Final Product

What students will submit as the final product of the activityA physical scaled model, a detailed construction plan, a final report explaining the mathematical concepts used and the design choices made, and documentation of the building process.

Alignment

How this activity aligns with the learning objectives & standardsIntegrates all standards (6.NS.A.1, 6.EE.B.6, 6.EE.B.8) by requiring students to compute quotients of fractions, use variables to represent numbers, write and interpret inequalities, and apply these concepts to a real-world design problem.
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Rubric & Reflection

Portfolio Rubric

Grading criteria for assessing the overall project portfolio

Scale Model Design Rubric

Category 1

Mathematical Accuracy

Assesses the correctness of calculations and the appropriate use of mathematical concepts related to fraction division, algebraic expressions, and inequalities.
Criterion 1

Fraction Division

Accuracy in calculating scaled dimensions using fraction division.

Exemplary
4 Points

All scaled dimensions are calculated correctly using fraction division, demonstrating a deep understanding of the concept. Calculations are clearly presented and explained.

Proficient
3 Points

Scaled dimensions are mostly calculated correctly using fraction division, with only minor errors. The understanding of the concept is evident.

Developing
2 Points

Scaled dimensions are calculated with some errors, indicating a partial understanding of fraction division. Calculations may be unclear or incomplete.

Beginning
1 Points

Scaled dimensions are calculated incorrectly or not attempted, demonstrating a limited understanding of fraction division.

Criterion 2

Algebraic Expressions

Appropriate use of variables and algebraic expressions to represent scaled dimensions.

Exemplary
4 Points

Variables are used effectively to represent original dimensions, and algebraic expressions accurately represent the scaled dimensions. Clear connection between expressions and numerical values is demonstrated.

Proficient
3 Points

Variables are used to represent original dimensions, and algebraic expressions mostly represent the scaled dimensions accurately. Minor errors may be present.

Developing
2 Points

Variables are used inconsistently, and algebraic expressions contain significant errors. Understanding of the relationship between variables and scaled dimensions is limited.

Beginning
1 Points

Variables are not used effectively, and algebraic expressions are absent or completely incorrect. Demonstrates a lack of understanding of algebraic representation.

Criterion 3

Inequalities

Correctly writing, solving, and interpreting inequalities to represent constraints.

Exemplary
4 Points

Inequalities are written and solved correctly, accurately representing the constraints on the model. The solution set is correctly represented on a number line, with a clear explanation of its meaning.

Proficient
3 Points

Inequalities are mostly written and solved correctly, with only minor errors. The solution set is represented on a number line, and its meaning is generally understood.

Developing
2 Points

Inequalities contain significant errors, and the solution set is either incorrect or not represented on a number line. Understanding of constraints and their representation is limited.

Beginning
1 Points

Inequalities are not written or solved correctly, and there is no representation of the solution set. Demonstrates a lack of understanding of inequalities and constraints.

Category 2

Design and Construction

Evaluates the planning, execution, and documentation of the scaled model construction.
Criterion 1

Model Plan

Clarity and completeness of the construction plan, including materials, tools, and steps.

Exemplary
4 Points

The construction plan is detailed, well-organized, and includes all necessary materials, tools, and steps. The plan clearly demonstrates how the model will meet the identified constraints.

Proficient
3 Points

The construction plan is mostly complete and includes most of the necessary materials, tools, and steps. The plan generally demonstrates how the model will meet the constraints.

Developing
2 Points

The construction plan is incomplete and lacks important details, such as specific materials or steps. The connection between the plan and the constraints is unclear.

Beginning
1 Points

The construction plan is minimal or absent, lacking essential information and failing to address the constraints.

Criterion 2

Model Construction

Quality of the scaled model, including accuracy, neatness, and adherence to the plan.

Exemplary
4 Points

The scaled model is accurately constructed, neatly finished, and closely adheres to the plan. The model effectively represents the chosen object and meets all constraints.

Proficient
3 Points

The scaled model is mostly accurately constructed and generally adheres to the plan. Minor inaccuracies or imperfections may be present.

Developing
2 Points

The scaled model contains significant inaccuracies or deviations from the plan. The construction may be messy or incomplete.

Beginning
1 Points

The scaled model is poorly constructed, bears little resemblance to the chosen object, and fails to meet the constraints.

Criterion 3

Documentation

Completeness and clarity of the final report, including explanation of design choices, mathematical concepts, and challenges encountered.

Exemplary
4 Points

The final report is comprehensive, well-written, and clearly explains all design choices, mathematical concepts, and challenges encountered during the construction process. The report demonstrates a deep understanding of the project and its underlying principles.

Proficient
3 Points

The final report is mostly complete and explains most of the design choices, mathematical concepts, and challenges encountered. The understanding of the project is evident.

Developing
2 Points

The final report is incomplete and lacks important details or explanations. The understanding of the project is limited.

Beginning
1 Points

The final report is minimal or absent, failing to address the design choices, mathematical concepts, or challenges encountered. Demonstrates a lack of understanding of the project.

Category 3

Application and Reasoning

Assesses how well the student applies mathematical concepts to the real-world problem and justifies their design choices with sound reasoning.
Criterion 1

Real-World Application

Effectiveness of applying mathematical concepts to the design of a real-world scaled model.

Exemplary
4 Points

Demonstrates a sophisticated understanding of how mathematical concepts can be applied to solve real-world design problems. The model is a creative and effective solution to the given problem.

Proficient
3 Points

Demonstrates a thorough understanding of how mathematical concepts can be applied to real-world design problems. The model is a functional solution to the given problem.

Developing
2 Points

Shows an emerging understanding of how mathematical concepts relate to real-world design problems. The model may have some flaws or inconsistencies.

Beginning
1 Points

Struggles to apply mathematical concepts to the design of a real-world scaled model. The model is incomplete or does not address the problem effectively.

Criterion 2

Justification of Choices

Soundness and clarity of reasoning used to justify design choices and mathematical approaches.

Exemplary
4 Points

Provides a clear and insightful rationale for all design choices, supported by strong mathematical reasoning. Demonstrates a deep understanding of the trade-offs and considerations involved in the design process.

Proficient
3 Points

Provides a reasonable rationale for most design choices, supported by mathematical reasoning. Demonstrates a good understanding of the design process.

Developing
2 Points

Provides a limited or unclear rationale for some design choices. Mathematical reasoning may be weak or missing.

Beginning
1 Points

Fails to provide a rationale for design choices, or the rationale is illogical or unsupported. Demonstrates a lack of understanding of the design process.

Reflection Prompts

End-of-project reflection questions to get students to think about their learning
Question 1

How did you apply your understanding of fractions to accurately scale down the dimensions of your chosen object in the final model?

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Question 2

Explain how you used algebraic expressions to represent and calculate the scaled dimensions of your model. How did using variables make the scaling process more efficient or adaptable?

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Question 3

To what extent do you agree with the statement: 'This project helped me understand how mathematical concepts can be applied to real-world design problems'?

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