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Created byAllison Fine
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Angles of Access: Designing ADA Compliant Ramps

Grade 10Math10 days
In this project, 10th-grade students assume the role of civil engineers tasked with designing ADA-compliant ramps for specific local building sites. Utilizing right triangle trigonometry and the properties of similar triangles, students calculate necessary ramp lengths and slopes while navigating real-world physical constraints. The experience culminates in the creation of scaled technical blueprints and 3D models, where students must mathematically justify their designs to ensure safe and equitable access for all.
TrigonometryADA ComplianceCivil EngineeringGeometric ModelingAccessibilityProportional Reasoning
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Inquiry Framework

Question Framework

Driving Question

The overarching question that guides the entire project.How can we, as civil engineers, use right triangle trigonometry and geometry to design an ADA-compliant ramp that ensures safe and equitable access within the physical constraints of a specific building site?

Essential Questions

Supporting questions that break down major concepts.
  • How can we use the properties of similar triangles and proportions to ensure a ramp maintains a consistent slope across different lengths?
  • How do trigonometric ratios help us determine the necessary ramp length and horizontal run based on the fixed height of a building's entrance?
  • In what ways do angles of elevation and depression influence the safety and usability of a design for people with different mobility needs?
  • How do engineers use right triangle geometry to navigate physical site constraints while strictly adhering to ADA (Americans with Disabilities Act) regulations?
  • Why is mathematical precision critical when designing infrastructure that must balance legal standards, physical space, and human safety?

Standards & Learning Goals

Learning Goals

By the end of this project, students will be able to:
  • Calculate unknown ramp lengths and horizontal runs using right triangle trigonometric ratios (sine, cosine, tangent).
  • Apply the concept of similar triangles and proportions to maintain a consistent slope across various ramp segments and landings.
  • Utilize angles of elevation and depression to ensure ramp designs meet specific ADA (Americans with Disabilities Act) safety requirements and site constraints.
  • Develop a scaled technical blueprint or 3D model that accurately represents the geometric dimensions of the proposed ramp.
  • Construct a mathematical argument to justify how a design balances physical site limitations with legal accessibility standards.

Common Core State Standards for Mathematics

CCSS.MATH.CONTENT.HSG.SRT.C.8
Primary
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.Reason: This is the core mathematical skill required to determine ramp lengths and slopes based on building heights.
CCSS.MATH.CONTENT.HSG.SRT.C.6
Primary
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.Reason: Students must understand the relationship between the slope (angle of elevation) and the ratios of the ramp's rise and run.
CCSS.MATH.CONTENT.HSG.SRT.B.5
Secondary
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.Reason: Students use similarity to ensure that regardless of the segment length, the slope remains proportional and compliant.
CCSS.MATH.CONTENT.HSG.MG.A.3
Supporting
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).Reason: The project specifically asks students to act as engineers designing a structure within the physical constraints of a building site and legal regulations.

Entry Events

Events that will be used to introduce the project to students

The Accessibility Audit: Hall of Shame

Students are presented with a 'Hall of Shame' photo gallery featuring local 'ramp fails'—inclines that are too steep or blocked. They are then handed a mock 'Compliance Audit' from the city, tasking them to use clinometers to prove why these ramps fail and to propose a trig-based redesign that meets the strict 1:12 ADA ratio.
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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

The Slope Sleuths: Decoding the 1:12 Ratio

Before designing, students must understand why the ADA 1:12 ratio is a geometric constant. In this activity, students analyze 'failed' ramps from the 'Hall of Shame' entry event. They will use the 1:12 ratio (1 inch of rise for every 12 inches of run) to determine the 'ideal' angle of elevation and compare it to the angles of non-compliant ramps. This establishes the mathematical foundation of slope as a trigonometric ratio (tangent) through visual exploration and calculation rather than formal proof.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Investigate the 'Hall of Shame' photos and identify three ramps that look too steep. Use the provided measurements of their rise and run to calculate their slopes as fractions and decimals to see how they deviate from the 1:12 standard.
2. Use the inverse tangent function [tan⁻¹(1/12)] to determine the maximum allowable angle of elevation (approximately 4.76 degrees) for a ramp to be ADA compliant.
3. On coordinate paper or digital graphing software, create a series of nested similar right triangles (e.g., 1 unit rise by 12 units run, 2 units rise by 24 units run). Use a protractor or software tool to verify that the angle of elevation remains identical for all triangles despite the change in scale.
4. Summarize the findings in the 'Ramp Audit Report' by explaining in plain language how the 1:12 ratio ensures a consistent, safe angle for users, regardless of how high the entrance is.

Final Product

What students will submit as the final product of the activityA 'Ramp Audit Report' that includes a comparison table of failed vs. compliant ratios, the calculation of the ideal ADA angle of elevation, and a visual similarity demonstration (graph) showing that a 1:12 ratio produces the same angle regardless of the ramp's size.

Alignment

How this activity aligns with the learning objectives & standardsAligns with HSG.SRT.C.6 (Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle) and HSG.SRT.B.5 (Use similarity criteria for triangles to solve problems).
Activity 2

Field Survey: Rise, Run, and Reality

Students transition from theory to the field. Using clinometers and measuring tapes, they visit their assigned building site to measure the total vertical 'rise' from the ground to the entrance. They must then calculate the required 'run' (horizontal distance) needed to meet ADA standards and use the Pythagorean Theorem to estimate the total length of the ramp surface (the hypotenuse).

Steps

Here is some basic scaffolding to help students complete the activity.
1. Use a clinometer and measuring tape to find the vertical height (rise) from the sidewalk to the threshold of the assigned building entrance.
2. Calculate the total horizontal run required using the 1:12 proportion (Run = Rise * 12).
3. Apply the Pythagorean Theorem (a² + b² = c²) to find the length of the actual ramp surface (hypotenuse) that will be built.
4. Identify and map physical site constraints (e.g., a tree, a sidewalk, or a fire hydrant) that might interfere with a straight ramp of that calculated length.

Final Product

What students will submit as the final product of the activityA 'Site Constraints Map' featuring a 2D geometric model of the building's entrance, labeled with measured rise, calculated required run, and estimated hypotenuse length.

Alignment

How this activity aligns with the learning objectives & standardsAligns with HSG.SRT.C.8 (Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems).
Activity 3

The Landing Logic: Navigating Site Constraints

ADA regulations state that a ramp cannot have a continuous run longer than 30 feet without a level landing (a 'rest' area). Since most ramps for high entrances will need to 'zig-zag' or turn, students must use similar triangles to design a multi-segment ramp. They will ensure that each segment maintains the exact same slope and angle as the whole, proving similarity between the smaller segments and the total theoretical ramp.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Divide your total calculated run from Activity 2 into segments no longer than 30 feet. Determine the number of level landings required.
2. Calculate the 'rise' for each individual segment. For example, if a segment is 15 feet long, what is the rise of just that section?
3. Draw a series of similar triangles representing each ramp segment. Label the angles of elevation for each to ensure they all match the ADA 'ideal' angle calculated in Activity 1.
4. Calculate the coordinates for each 'turn' or 'landing' in the design to ensure the ramp fits within the building's property lines.

Final Product

What students will submit as the final product of the activityA 'Segmented Ramp Schematic' showing the breakdown of the ramp into sections and landings, including a mathematical proof that the triangles formed by each segment are similar to the 1:12 master triangle.

Alignment

How this activity aligns with the learning objectives & standardsAligns with HSG.SRT.B.5 (Use congruence and similarity criteria to solve problems) and HSG.SRT.C.8 (Solve right triangles).
Activity 4

Trig Tech Blueprints: The Engineer's Draft

Now, students act as lead engineers to synthesize their data into a professional technical blueprint. They will use Sine, Cosine, and Tangent to verify every dimension of their design. This blueprint must include 'Angle of Depression' calculations from the top of the ramp to ensure safety for someone descending, and it must be drawn to a specific scale (e.g., 1/4 inch = 1 foot).

Steps

Here is some basic scaffolding to help students complete the activity.
1. Choose an appropriate scale for your drawing so the entire design fits on the provided drafting paper.
2. Draft the side profile of the ramp, using Sine and Cosine to double-check the lengths of the ramp surfaces based on your angles of elevation.
3. Calculate and label the angle of depression from the top landing to the bottom to verify descending safety.
4. Include a 'Calculation Key' on the side of the blueprint showing the Sine, Cosine, and Tangent equations used to verify the design's accuracy.

Final Product

What students will submit as the final product of the activityA 'Scaled Technical Blueprint' of the final ramp design, featuring top-down and side-profile views with all angles, lengths, and proportions clearly labeled and verified with trigonometric calculations.

Alignment

How this activity aligns with the learning objectives & standardsAligns with HSG.MG.A.3 (Apply geometric methods to solve design problems) and HSG.SRT.C.8 (Use trig ratios to solve right triangles).
Activity 5

The Engineer's Prototype: Building for Equity

In this final synthesis activity, students move from 2D drafting to 3D construction by building a precise scale model of their ramp using cardstock. This physical prototype serves as the centerpiece for their final presentation to the 'Compliance Board.' Students must ensure their physical model matches their blueprint dimensions exactly, providing a tangible proof of concept that addresses their specific site constraints. To conclude the project, they will present their model alongside a written Engineering Brief that justifies their design choices using trigonometric data and explains how their solution ensures equitable access.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Translate your blueprint measurements into physical components. Use cardstock to cut out the ramp surfaces (hypotenuses), side walls, and landings, ensuring every piece is scaled accurately (e.g., 1/4 inch = 1 foot).
2. Assemble the cardstock model. As you build, use a protractor or clinometer to verify that the physical angle of elevation on your model matches your trigonometric calculations (ideally 4.76 degrees for a 1:12 slope).
3. Prepare the 'Math Justification' section of your Engineering Brief. Prove the ramp's compliance by citing the sine, cosine, and tangent ratios used during the build and explain how you navigated specific site constraints (e.g., switchbacks or wrap-around landings).
4. Present your 3D model and Engineering Brief to the class. Demonstrate how the ramp fits the building's entrance and defend your mathematical precision during a Q&A session with the 'Compliance Board.'

Final Product

What students will submit as the final product of the activityA 3D physical scale model of the ramp made from cardstock, a written 'ADA Compliance Engineering Brief,' and an oral presentation defending the mathematical and structural integrity of the design.

Alignment

How this activity aligns with the learning objectives & standardsAligns with HSG.MG.A.3 (Apply geometric methods to solve design problems; work with physical constraints) and HSG.SRT.C.8 (Use trigonometric ratios to solve right triangles).
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Rubric & Reflection

Portfolio Rubric

Grading criteria for assessing the overall project portfolio

Civil Engineering: ADA Ramp Design & Trig Synthesis Rubric

Category 1

Mathematical Modeling & Application (Standards: HSG.SRT.C.6, HSG.SRT.C.8, HSG.SRT.B.5)

Focuses on the core mathematical skills of trigonometry, geometry, and proportional reasoning as applied to the ramp design.
Criterion 1

Trigonometric Accuracy & Solving Right Triangles

Evaluation of the student's ability to use sine, cosine, tangent, and their inverses to calculate ramp lengths, horizontal runs, and angles of elevation/depression.

Exemplary
4 Points

Calculations are flawless across all ramp segments. Student independently uses inverse tangent to find the ADA angle (4.76°) and applies sine/cosine to verify all hypotenuses and runs with 100% accuracy. Angles of depression are correctly calculated to ensure descending safety.

Proficient
3 Points

Calculations are mostly accurate with minor rounding errors. Student correctly identifies which trigonometric ratios to use for most parts of the ramp design, including the primary angle of elevation and basic run calculations.

Developing
2 Points

Calculations show emerging understanding but contain inconsistent errors. Student can solve for one part of a triangle (e.g., the run) but struggles with more complex applications like inverse functions or angles of depression.

Beginning
1 Points

Calculations are incomplete or frequently incorrect. Student struggles to identify the correct trigonometric ratio for a given side or angle and requires significant support to apply the Pythagorean Theorem.

Criterion 2

Similarity and Proportional Reasoning

Assessment of the student's ability to use similarity and proportions (1:12 ratio) to maintain a consistent slope across multiple segments and landings.

Exemplary
4 Points

Independently demonstrates through consistent calculation and geometric reasoning that all ramp segments form similar triangles with the 1:12 master ratio. Clearly shows that the angle of elevation remains constant regardless of the segment length or total height.

Proficient
3 Points

Correctly applies the 1:12 ratio to all segments and accurately calculates the required run for the measured rise. Identifies that segments must be proportional to maintain a legal slope.

Developing
2 Points

Applies the 1:12 ratio to the total ramp but struggles to maintain proportionality when breaking the ramp into smaller segments or landings. Explanation of similarity is partial or lacks clarity.

Beginning
1 Points

Fails to apply the 1:12 ratio correctly. Does not demonstrate an understanding of how similar triangles relate to a consistent slope or angle of elevation.

Category 2

Engineering Design & Physical Modeling (Standard: HSG.MG.A.3)

Focuses on the student's ability to translate mathematical theory into technical drawings and physical models within real-world constraints.
Criterion 1

Technical Drafting & Scaling

Assessment of the technical blueprint's accuracy, scale, and inclusion of necessary engineering details (rise, run, angles, site constraints).

Exemplary
4 Points

Blueprint is of professional quality, drawn to a precise scale (e.g., 1/4" = 1'). Includes comprehensive labels for all geometric components and a 'Calculation Key' that flawlessly documents the trig equations used for verification.

Proficient
3 Points

Blueprint is clearly drawn to scale with all major dimensions (rise, run, hypotenuse) and angles labeled. Site constraints are identified and integrated into the design reasonably well.

Developing
2 Points

Blueprint is attempted but contains scaling errors. Some labels are missing or unclear, and the side profile/top-down views do not align perfectly with the mathematical data.

Beginning
1 Points

Blueprint is disorganized, not to scale, or missing essential geometric information. Does not provide a clear visual representation of the proposed ramp design.

Criterion 2

3D Prototyping & Constraint Navigation

Evaluation of the physical 3D cardstock model's fidelity to the mathematical design and the student's ability to navigate site-specific obstacles.

Exemplary
4 Points

3D model is a perfect physical manifestation of the blueprint. Student innovatively navigates complex site constraints (e.g., switchbacks around a tree) while maintaining strict ADA compliance. Physical angles match trig calculations exactly when measured.

Proficient
3 Points

3D model accurately represents the 2D design and accounts for basic site constraints. The model is structurally sound and reflects the calculated proportions and angles.

Developing
2 Points

3D model is completed but shows inconsistencies with the blueprint. Construction may be imprecise, making it difficult to verify if the physical angle of elevation matches the mathematical design.

Beginning
1 Points

3D model is incomplete or does not follow the calculated dimensions. Site constraints are ignored, and the model does not demonstrate a functional understanding of the design problem.

Category 3

Communication & Advocacy

Focuses on the synthesis of the project through written documentation and oral presentation, emphasizing the 'why' behind the design.
Criterion 1

Mathematical Justification & Oral Defense

Evaluation of the student's ability to explain their design choices, mathematical reasoning, and how their ramp ensures equitable access.

Exemplary
4 Points

The Engineering Brief and presentation provide a compelling, data-driven argument for the design. Student uses precise mathematical terminology and confidently defends their design against 'Compliance Board' questions using trig evidence. Explicitly connects math to the social goal of equity.

Proficient
3 Points

The Engineering Brief clearly explains the calculations and design choices. Presentation is organized and uses appropriate mathematical language to justify how the ramp meets ADA standards.

Developing
2 Points

The Engineering Brief describes the project but lacks deep mathematical justification. Presentation is basic, and the student may struggle to explain why certain trigonometric ratios or design choices were made.

Beginning
1 Points

The Engineering Brief is missing or lacks detail. Presentation is unorganized, and the student cannot explain the mathematical basis for their ramp design.

Reflection Prompts

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

How did the concept of similar triangles help you ensure that your ramp remained ADA-compliant even when you had to break it into multiple segments or landings?

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Question 2

On a scale of 1 to 5, how challenging was it to balance strict ADA mathematical requirements with the physical constraints (trees, sidewalks, etc.) of your specific building site?

Scale
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Question 3

Which mathematical tool did you find most critical when transitioning from your initial site measurements to your final 2D blueprint?

Multiple choice
Required
Options
Tangent / Inverse Tangent (calculating angles from rise/run)
Sine / Cosine (verifying ramp surface lengths)
Pythagorean Theorem (finding the hypotenuse)
Geometric Proportions (maintaining the 1:12 ratio across segments)
Question 4

As a 'civil engineer,' how has this project changed your perspective on the importance of mathematical precision in public infrastructure? Why is a 0.5-degree error more than just a 'math mistake' in this context?

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