Equation Architect Slides Project: Equation Architect
MATH BY DESIGN
Solving Two-Step Equations with Rational Numbers
THE MISSION
Isolate
Master the two-step process to isolate the variable.
Calculate
Navigate decimals, fractions, and negative signs with precision.
Construct
Use your solutions to build a custom character blueprint.
THE TWO-STEP PROTOCOL
1
Undo Add/Sub
Perform the inverse operation on the constant term.
2
Undo Mult/Div
Isolate the variable by removing the coefficient.
Architect's Rule
"What you do to one side of the equation, you MUST do to the other."
EXAMPLE 01: INTEGERS
STATUS: STABLE
\( 3x + 12 = 30 \)
Step 01
Subtract 12: \( 3x = 18 \)
Step 02
Divide by 3: \( x = 6 \)
Verification
\( 3(6) + 12 = 30 \)
EXAMPLE 02: DECIMALS
STATUS: PRECISION
\( 0.5x - 4 = 1 \)
+ 4 + 4
\( 0.5x = 5 \)
\(\div 0.5\) \(\div 0.5\)
\( x = 10 \)
Design Tip
Dividing by 0.5 is the same as multiplying by 2.
Keep your foundation steady—align your decimals!
EXAMPLE 03: FRACTIONS
STATUS: STRUCTURAL
\( \frac{1}{4}x + 10 = 8 \)
1. Clear the Constant
Subtract 10: \( \frac{1}{4}x = -2 \)
2. Clear the Fraction
Multiply by 4: \( x = -8 \)
Structural Warning
Negative results are common. Keep the sign!
CONSTRUCTION PHASE
1
Solve the 10 equations on your worksheet.
2
Check the Multiple-Choice Key for your answer.
3
Draw your persona in the Drawing Zone.
Self-Correction Mode
Equation Architect Worksheet IDENTITY ARCHITECT
Mathematical Design Protocol / Phase 1
Architect: _________________
Date: _______
Solve each equation below. Show all structural work. Your results for x will be used to decode your character's facial features on Page 2.
01. Eyes
\( 3x + 12 = 30 \)
02. Nose
\( 0.5x - 4 = 1 \)
03. Mouth
\( \frac{1}{4}x + 10 = 8 \)
04. Hair
\( -2x + 15 = 1 \)
05. Ears
\( 5x - 2.5 = 12.5 \)
06. Brows
\( 4x - 18 = 6 \)
Identity Architect: Phase 2
PAGE 02 / 02
07. Cheeks
\( 1.5x + 7 = 10 \)
08. Neck
\( -3x - 10 = -25 \)
09. Hat
\( 0.8x + 2.6 = 5 \)
10. Back
\( \frac{x}{3} + 1 = 4 \)
Selection Key
Decode Specs
UNIT CHOICE A CHOICE B CHOICE C 01. Eyes x = 6 (Glasses) x = 14 (Round) x = 4 (Monocle) 02. Nose x = 6 (Circle) x = 10 (Tri Nose) x = -6 (Pig) 03. Mouth x = -8 (Smile) x = 8 (Frown) x = -0.5 (Moustache) 04. Hair x = 8 (Curly) x = -8 (Bald) x = 7 (Spiky Hair) 05. Ears x = 2 (Small) x = -3 (Robot) x = 3 (Pointy) 06. Brows x = 6 (Bushy) x = 3 (Thin) x = -6 (Uni-brow) 07. Cheeks x = 11.3 (Rosy) x = 2 (Freckles) x = -2 (Bandage) 08. Neck x = 5 (Bowtie) x = -5 (Scarf) x = 11.6 (Necklace) 09. Hat x = -3 (Baseball) x = 9.5 (Top Hat) x = 3 (Beanie) 10. Back x = 9 (Stars) x = 15 (Bricks) x = 1 (Dots)
Blueprint Zone
Draw Persona Below
Equation Architect Teacher Guide TEACHER PROTOCOL
Lesson: Equation Architect
Duration
55 MIN
Learning Targets
Isolate variables using inverse operations in exactly two steps.
Apply operations to rational numbers (integers, decimals, and fractions).
Identify and correct common errors through multiple-choice verification.
distractor logic
Common Errors
Distractors in the Feature Key include sign errors (e.g., mouth $x=8$), reciprocal errors (e.g., mouth $x=-0.5$), and simple arithmetic slips.
SPECIFICATION KEY (ANSWERS)
01. Eyes x = 6 (A)
06. Brows x = 6 (A)
02. Nose x = 10 (B)
07. Cheeks x = 2 (B)
03. Mouth x = -8 (A)
08. Neck x = 5 (A)
04. Hair x = 7 (C)
09. Hat x = 3 (C)
05. Ears x = 3 (C)
10. Back x = 9 (A)
Teacher Note: Ensure students justify why the distractors are incorrect for deeper conceptual reinforcement.
Launch
10m
Construct
30m
Reveal
15m
Equation Architect Answer Key ANSWER KEY
Identity Architect / Evaluation Protocol
Teacher Resource
Quick Reveal Table
UNIT ANSWER CHOICE 01. Eyes x = 6 Choice A 02. Nose x = 10 Choice B 03. Mouth x = -8 Choice A 04. Hair x = 7 Choice C 05. Ears x = 3 Choice C
Quick Reveal Table
UNIT ANSWER CHOICE 06. Brows x = 6 Choice A 07. Cheeks x = 2 Choice B 08. Neck x = 5 Choice A 09. Hat x = 3 Choice C 10. Back x = 9 Choice A
Step-by-Step Structural Analysis
01. 3x + 12 = 30
1. Sub 12: 3x = 18
2. Div 3: x = 6
Choice A (Glasses)
02. 0.5x - 4 = 1
1. Add 4: 0.5x = 5
2. Div 0.5: x = 10
Choice B (Tri Nose)
03. 1/4x + 10 = 8
1. Sub 10: 1/4x = -2
2. Mult 4: x = -8
Choice A (Smile)
04. -2x + 15 = 1
1. Sub 15: -2x = -14
2. Div -2: x = 7
Choice C (Spiky)
05. 5x - 2.5 = 12.5
1. Add 2.5: 5x = 15
2. Div 5: x = 3
Choice C (Pointy)
06. 4x - 18 = 6
1. Add 18: 4x = 24
2. Div 4: x = 6
Choice A (Bushy)
07. 1.5x + 7 = 10
1. Sub 7: 1.5x = 3
2. Div 1.5: x = 2
Choice B (Freckles)
08. -3x - 10 = -25
1. Add 10: -3x = -15
2. Div -3: x = 5
Choice A (Bowtie)
09. 0.8x + 2.6 = 5
1. Sub 2.6: 0.8x = 2.4
2. Div 0.8: x = 3
Choice C (Beanie)
10. x/3 + 1 = 4
1. Sub 1: x/3 = 3
2. Mult 3: x = 9
Geometry Mastery Choice Board GEOMETRY MASTERY LAB
Final Evaluation: MA.7.G.A & B
Duration
180 MIN
Researcher Protocol
Demonstrate your geometric mastery by choosing ONE specialized research path. Each path is a robust, 3-hour assessment designed to test your technical drawing, precise calculation, and algebraic logic. Review the options below and select your assignment.
Project
01. The Architect
Scale floor-plan design (1cm:4ft) for a luxury suite. Includes composite area budgeting and circular balcony specs.
Construction
02. The Artist
Technical mural construction (4 triangle types) using tools. Features hidden algebraic "Angle Ciphers" within the art.
Lab Report
03. The Slicer
3D imaging lab. Visualize and name 2D cross-sections of prisms and pyramids. Includes the "Slant Challenge" visualization.
Design Task
04. The Chef
Business analytics for a pizzeria. Circle mechanics (Area/Circumference) and 3D logistics for packaging efficiency.
05. THE SCHOLAR
The comprehensive gauntlet. A formal 12-problem assessment covering all MA geometry standards in high rigor.
Type: Formal Test
PROTOCOL V2.2 // MASTERY HUB
© 2026 GEOMETRY LABS
Architect Scale Project Assignment PATH 01: THE ARCHITECT
Project: Urban Living Studio Scale Drawing
Standards: 7.G.A.1, 7.G.B.6
Duration: 180 Minutes
Project Brief
You are designing a luxury 40 ft by 32 ft studio. You must produce a precise scale drawing where 1 cm = 4 ft . Your client requires the following:
1. The Lounge: L-Shape (Min 40% area)
2. The Kitchen: Rectangle (12 ft x 16 ft actual)
3. The Balcony: Semi-circle (Diameter = 16 ft)
Conversion Checklist
<table class="w-full text-[10px] font-mono border-collapse"><tbody><tr class="bg-blue-50 font-bold"><td class="p-2 border">Actual Dim</td><td class="p-2 border">Drawing (cm)</td></tr><tr><td class="p-2 border">40 ft (Length)</td><td class="p-2 border">10.0 cm</td></tr><tr><td class="p-2 border">32 ft (Width)</td><td class="p-2 border">8.0 cm</td></tr><tr><td class="p-2 border">12 ft (Kitchen)</td><td class="p-2 border">3.0 cm</td></tr><tr><td class="p-2 border">16 ft (Kitchen)</td><td class="p-2 border">4.0 cm</td></tr><tr><td class="p-2 border">16 ft (Diameter)</td><td class="p-2 border">4.0 cm</td></tr></tbody></table>
Teacher Note: Conversions provided for self-check validation.
Architectural Canvas // 1 Box = 1 CM
Calculation Log: Path 01
1. Composite Area Decomp (7.G.B.6)
Calculate total actual area of the suite. (L x W). Then decompose into zones.
Show Algebraic Work Here...
2. Circular Balcony Specs (7.G.B.4)
Radius = 8 ft. Use π ≈ 3.14. Calculate Area (divided by 2) and Arc Length.
Show Algebraic Work Here...
Flooring Budget
Hardwood flooring is priced at $12.50 / sq ft.
Total Cost Estimate Calculation:
Total: $____________
Student-Facing Rubric
Criteria Expert (4) Practitioner (3) Novice (2/1) Scale Drawing (7.G.A.1) Drawing matches 1cm:4ft perfectly. All zones placed correctly on canvas. Most dimensions accurate; 1-2 minor conversion or plotting errors. Significant scaling errors; drawing does not represent intended dimensions. Polygon Area (7.G.B.6) Total suite area (1280 sq ft) and room areas calculated flawlessly with clear steps. Correct strategy; minor arithmetic errors in sub-areas or budget. Failed to correctly identify or calculate areas of suite parts. Circle Logic (7.G.B.4)
Artist Angle Mural Assignment PATH 02: THE ARTIST
Project: Geometric Mural & Angle Ciphers
Standards: 7.G.A.2, 7.G.B.5
Duration: 180 Minutes
Phase 1: Mural Specs
Construct 4 specific geometric elements into your mural design. You must use a ruler and protractor.
1. Isosceles: 8cm sides, 40° vertex.
2. Scalene: sides 5cm, 9cm, 12cm.
3. Right: base 10cm, 30° angle.
4. Intersecting "X" for vertical ciphers.
Phase 2: Angle Ciphers
Label these intersections in your design:
Cipher X (Vertical): 4x + 10 and 90°
Cipher Y (Supp): 2x - 5 and 3x + 10
Cipher Z (Comp): x + 20 and 2x
Technical Mural Canvas
MEASURE WITH PRECISION // LEAVE CONSTRUCTION MARKS
Angle Cipher Proof Sheet
UNIT: 7.G.B.5
Cipher X: Vertical Pair
Set 4x + 10 = 90. Solve for x.
Equation Solving Work:
Cipher Y: Supplementary Sum
Set (2x - 5) + (3x + 10) = 180. Solve for x.
Equation Solving Work:
Cipher Z: Complementary Sum
Set (x + 20) + 2x = 90. Solve for x.
Equation Solving Work:
Student-Facing Rubric
Criteria Expert (4) Practitioner (3) Novice (2/1) Technical Construction (7.G.A.2) All 3 triangles constructed with 100% precision using tools. Protractor marks are clear. All triangles present; minor errors (±2mm or ±2°) in construction. Significant construction errors; triangles drawn freehand or miss specs. Angle Logic (7.G.B.5) Cipher proofs solve for x perfectly. All relationships correctly modeled by equations. Relationships correctly identified; minor slips in the isolation of x. Incorrectly modeled relationships or missing algebraic work. Algebraic Rigor Shows clear, step-by-step proof for solving all three ciphers. Verification shown. Solving is clear; missing minor verification or labeling. Algebraic work is missing or unorganized. Visual Design Mural is professional, striking, and geometrically sound. Technique is impeccable. Mural is organized and follows all prompt requirements. Mural is messy, incomplete, or lacks evidence of tool use.
Mastery Score:
Slicer Cross Section Lab Assignment PATH 03: THE SLICER
Lab Report: 3D Imaging & Internal Geometry
Standards: 7.G.A.3, 7.G.B.6
Duration: 180 Minutes
Lab Protocol
As a technician, you must visualize the 2D intersections of planes and 3D solids. Build models (clay or digital) and document your "scans" below.
Scan 1: Prismatic Imaging
A. Rect. Prism (Horizontal)
Slice parallel to the base.
Shape: _________________
B. Rect. Prism (Vertical)
Slice perpendicular to base.
Shape: _________________
Scan 2: Pyramidal Analysis
A. Pyramid (Horizontal)
Slice halfway up the pyramid.
Shape: _________________
B. Pyramid (Apex)
Vertical slice through apex.
Shape: _________________
Imaging Log: Part 2
Lab ID: 7.G.A.3
Scan 3: Triangular Prism Analysis
Build a Right Triangular Prism . Identify the shapes created by different scans.
A. Transverse Scan
Parallel to the triangular base.
Result: _________________
B. Longitudinal Scan
Perpendicular to the base.
Result: _________________
The Slant Challenge
Visualization: Slice a Cube diagonally from one top corner to the opposite bottom corner. Identify the shape and provide a reasoned argument below.
Internal Sketch Area
Reasoning Log:
Student-Facing Rubric
Criteria Expert (4) Practitioner (3) Novice (2/1) Cross-Section ID (7.G.A.3) All 2D cross-sections (Scans 1, 2, 3) identified and drawn with perfect geometric detail. Most shapes identified; minor inaccuracies in naming or drawing. Fails to correctly name or draw the resulting 2D shapes. Visualization Proof Slant Challenge is solved with advanced 3D perspective and mathematical reasoning. Challenge answered correctly; reasoning is mostly sound but lacks depth. Visualization is incorrect or lacks supporting evidence. Geometric Vocabulary Uses precise terms (parallel, perpendicular, plane, transverse) throughout the report. Uses basic geometric terms correctly. Major terminology errors or missing vocabulary.
Chef Circle Pizzeria Assignment PATH 04: THE CHEF
Business Analytics: Circle Pizzeria Design
Standards: 7.G.B.4, 7.G.B.6
Duration: 180 Minutes
Culinary Logic
Master circle geometry to optimize topping costs and packaging efficiency. Signature Diameter = 14 inches . (Use π ≈ 3.14).
Phase 1: The "Pi-thon" Specs
A. Total Surface Area
Calculate Area (A = π r²).
Show Steps:
B. Crust Circumference
Calculate Circumference (C = π d).
Show Steps:
Fractional Coverage
If exactly 1/8 (one-eighth) of the pizza is covered in spinach, what is the square-inch area of the spinach? Show the fractional area calculation.
Proof: Area / 8 = ______________
Logistics Analytics: Path 04
Phase 2: Box Design Net
Design a box for your pizza (16in x 16in x 2in). Construct the FLAT NET of the rectangular prism box below. Label all face dimensions.
Box Net Canvas
Total Surface Area
Sum of 6 face areas.
Show Math...
Total Volume
V = Bh or l*w*h.
Show Math...
The Efficiency Challenge
Which business model uses more Storage Volume : One "Jumbo" box (20 x 20 x 2) or four "Individual" boxes (10 x 10 x 2 each)? Use volume formulas to provide a mathematical proof and a final business verdict.
Comparative Calculation Area:
Verdict:
Student Rubric
Criteria Expert (4) Practitioner (3) Novice (2/1) Circle Mechanics (7.G.B.4) Flawless calculation of Area, Circumference, and Fractional Areas. Pi (3.14) used correctly. Most formulas correct; minor arithmetic slips in multiplication. Major confusion between diameter/radius or area/circumference. 3D Logistics (7.G.B.6) Box Net is drawn perfectly to scale. Surface Area and Volume are 100% accurate. Net is recognizable but contains minor scaling errors. Correct SA/Vol strategies. Incorrect calculation of SA or Volume; Net is missing or disproportionate. Comparative Logic Efficiency Challenge is solved with advanced comparative math and a logical business verdict. Calculations for both box types are correct; minor errors in comparison logic.
Scholar Geometry Gauntlet Exam Assignment PATH 05: THE SCHOLAR
Comprehensive Mastery Exam: Geometry Gauntlet
Standard Focus: ALL (MA.7.G.A & B)
Duration: 180 Minutes
"The Scholar path is a formal test designed to challenge the limits of your geometric knowledge. Rigorous calculation, precise drawing, and clear algebraic reasoning are required."
I. The Architect's Foundation (7.G.A.1)
Q1. Scale Conversions:
A regional park scale drawing uses 1 inch = 20 yards . If the actual park is 100 yards by 60 yards, calculate drawing dimensions and the scale factor. Then calculate actual area (yards²) and drawing area (inches²).
Show Algebraic Steps Here...
Part II: Precise Construction (7.G.A.2)
Q2. Technical Blueprint:
Using a metric ruler and protractor, construct a triangle with side lengths 6 cm and 8 cm , and an included angle of exactly 50° . Label all sides and angles. Leave your construction marks visible for the examiners.
Technical Construction Grid
Part III: Algebra & Pi Pioneer
III. The Cipher Specialist (7.G.B.5)
Q3. Supplementary Puzzles:
Angles A and B are supplementary. A = 3x - 10 and B = 2x + 20. Solve for x and each angle.
Proof Log:
Q4. Intersecting Lines:
Vertical angles: One is 5y, the other is 100°. Solve for y and find the adjacent angle.
Proof Log:
IV. Pi Pioneer (7.G.B.4)
Q5. Exact Circle Mechanics:
A circle has a radius of 10 meters. Calculate EXACT Area and Circumference (with π).
Proof Log:
Q6. Reverse Engineering:
Stage Area = 64π sq ft. What is the diameter? Show reverse algebraic steps.
Proof Log:
Part IV: 3D Gauntlet
Q7. Cross-Sections
A cube is sliced horizontally. Name the 2D shape. Area if side = 5cm?
Q8. Prism Volume
Rt Triangular Prism: base sides 3,4,5. Prism height = 12. Volume?
Q9. Surface Area
Total SA of a cube with side length 4.5 cm? Show formula.
V. PERFORMANCE CHALLENGE
Q10. Composite Prism gauntlet:
Join Block A (2x2x6) and Block B (4x2x2) into an "L" shape. Find total Volume and Surface Area.
Sketch Area
Final Proof:
END OF EXAM // SCHOLAR PATHWAY
SCORE: _________ / 10
Geometry Mastery Teacher Keys Material TEACHER EVALUATION KEYS
Geometry Mastery Solutions (MA.7.G)
PATH 01
Conversions (1cm : 4ft)
Length: 40/4 = 10cm; Width: 32/4 = 8cm
Area & Budget Results
Total Area: 1,280 sq ft
PATH 02
Angle Cipher Keys
X (Vert): 4x+10=90 → x=20
Verify Murals
Use metric ruler. Cipher x=35 means angles are 65° and 115°.
PATH 03
Cross-Section ID Key
1A (Prism Horiz): Rectangle
1B (Prism Vert): Rectangle
2A (Pyramid Horiz): Rectangle
2B (Pyramid Apex): Triangle
3A (Tri-Prism Horiz): Triangle
3B (Tri-Prism Vert): Rectangle
Slant Challenge Result
Shape: Trapezoid or Rectangle (plane through 4 faces).
PATH 04
Circle Data (d=14, r=7)
Area: 153.86 in²
Efficiency Verdict
Jumbo (20x20x2) = 800 in³.
PATH 05: THE SCHOLAR (EXAM KEY)
Q1 Scaling: Dwg: 5in x 3in. Actual Area: 6,000 yd². Dwg Area: 15 in².
Q2 (Triangle): Verify base angles of 65° and 65° (if isosceles attempt) or just side lengths 6, 8 with 50°.
Q3 Supp: 5x+10=180 → x=34. A=92°; B=88°.
Q4 Vert: 5y=100 → y=20. Adj = 80°.
Q5 Pi: C=20π (~62.8m). A=100π (~314m²).
Q6 Stage: r²=64 → r=8. Diameter = 16 ft.
Q7 Cube: Shape: Square. Area: 25 cm².
Q8 Prism: Base Area 6. Vol = 6 * 12 = 72 units³.
Q9 Cube SA: 6 * (4.5 * 4.5) = 121.5 cm².
Q10 COMPOSITE GAUNTLET
Total Vol: 24 (A) + 16 (B) = 40 units³.
Generic Project Rubric RUBRIC
Generic Project Assessment Scale
Student Name: _________________
Date: _______
Categories 4 Points 3 Points 2 Points 1 Point Completed Project All pages of project complete and color picture included. Most pages of project complete and color picture included. Some pages of project complete and picture included. One page or less of project complete and/ or no picture. Math All problems have been written on scratch paper and are correct. Most problems written on scratch paper and correct. Some problems written on scratch paper and correct. No proof of work provided. Neatness and Organization Project was extremely neat, organized, and easy to read. Project was neat, organized, and could be read. Project was somewhat neat and organized. Project was messy and not organized. On-Task On-task everyday. On-task most days. On-task some days. Did not stay on task. Followed Directions Followed all directions. Followed most directions. Followed some directions. Did not follow directions.
Final Grade:
/ 20
Rainbow Probability Student Lab RAINBOW PROBABILITY LAB
7th Grade Math / Probability Investigations (7.SP.C)
Student Name: _________________
Date: _______
Your Mission: Can we use math to predict the future? Today, you will use a real bag of Skittles to collect data, build probability models, run simulations to analyze complex compound events, and see how closely your experimental results match theoretical expectations.
Phase 1: Organizing Your Sample Space (7.SP.C.7)
Open your bag of Skittles. Sort your candies into the five possible color outcomes: Red, Orange, Yellow, Green, and Purple . Count how many of each color you have and write the numbers below.
Red (R)
Count: _____
Orange (O)
Count: _____
Yellow (Y)
Count: _____
Green (G)
Count: _____
Purple (P)
Count: _____
Total Candies in Your Bag (Total Sample Size) = ____________ Sum of all individual probabilities = 1.00 (or 100%)
Phase 2: Simple Probability Predictions (7.SP.C.5, 7.SP.C.6)
Q1. Probability of drawing a Non-Red Candy:
Using your bag's counts, calculate the probability of picking a candy that is not Red . Write your answer as a fraction, decimal, and percentage.
Q2. Law of Large Numbers Investigation:
If we combined the candy counts from 10,000 classrooms, would our experimental results get closer to or further from the manufacturer's target distribution? Explain why.
RAINBOW PROBABILITY LAB // PHASE 1 & 2
PAGE 01 / 02
Compound Gauntlet
PAGE 02 / 02
Phase 3: Compound Probability with Replacement (7.SP.C.8.a)
You draw a candy, record the color, put it back in the bag (replacement) , and draw a second candy. Use your Phase 1 counts to construct a tree diagram for Red (R) and Purple (P) draws below.
1. Draw Tree Diagram (With Replacement)
2. Calculate: Probability of Red, then Purple
P(Red, then Purple) = _________________
Phase 4: Compound Probability without Replacement (7.SP.C.8.c)
You draw a candy, eat it (no replacement ), and draw a second candy. How does this change the total number of options in the bag? Calculate the probability of drawing Red, then Purple under this new condition.
Show Your Step-by-Step Multiplication:
Evaluation Rubric
Rainbow Probability Teacher Key TEACHER KEY & FACILITATION GUIDE
Rainbow Probability Lab / MA.7.SP.C Evaluation
Model Key
Grading Framework
Because student bags of Skittles will contain slightly different total counts and color distributions, this key utilizes a standard model distribution of exactly N = 50 candies to demonstrate correct mathematical steps. Grade students based on the consistency of their equations relative to their unique counted total.
Model Distribution (N = 50)
Red: 10
P(R) = 0.20
Orange: 12
P(O) = 0.24
Yellow: 8
P(Y) = 0.16
Green: 11
P(G) = 0.22
Purple: 9
P(P) = 0.18
Solution Matrix
Q1. P(Non-Red Candy) Solution:
Using model data: \( N = 50 \), Red = 10. Non-Red = 40. Formula: \( P(\text{Not Red}) = \frac{40}{50} = \frac{4}{5} \).Decimal: 0.80 | Percentage: 80%
Q2. Law of Large Numbers Explanation:
As sample size increases toward infinity, the experimental probability will converge with the theoretical probability. Minor individual variations (such as a single packet with no green candies) will be smoothed out by the massive sample size.
Phase 3. P(Red, then Purple) - With Replacement:
Theoretical rate: \( P(\text{Red}) = \frac{10}{50} = 0.2 \) | \( P(\text{Purple}) = \frac{9}{50} = 0.18 \). Independent events.
Phase 4. P(Red, then Purple) - Without Replacement:
Dependent events. First draw: \( P(\text{Red}) = \frac{10}{50} \). Second draw: total sample decreases to 49, purple remains 9.
RAINBOW PROBABILITY LAB // EVALUATION MASTER KEY
PAGE 01 / 01
Probability Mastery Choice Board PROBABILITY MASTERY LAB
Massachusetts Standards: 7.SP.C.5 — 7.SP.C.8
Duration
180 MIN
Researcher Protocol
Demonstrate your probability and statistical mastery by choosing ONE specialized research path. Each path is a robust 3-hour engagement. Review the options below and select your assignment packet from the laboratory archives.
Project
01. The Casino Mogul
Design a game and test its fairness. Compare theoretical probability to 100 actual trials to analyze outcome stability.
Analysis
02. The Data Detective
Research real-world non-uniform events. Compare observed frequencies to uniform models and justify the results.
Mapping
03. The Tree Architect
Master complex sample spaces. Use tree diagrams and tables to identify exact compound probabilities for 3-stage events.
Simulation
04. The Simulator
Design and run an 80-trial simulation. Predict frequencies and analyze how sample size impacts accuracy.
05. THE SCHOLAR
The comprehensive gauntlet. A formal 15-problem assessment covering all MA probability standards in high rigor.
Type: Formal Test
PROTOCOL V1.4 // PROBABILITY LABS
© 2026 PROBABILITY LABS
Casino Mogul Project Assignment PATH 01: THE CASINO MOGUL
Project: "Fair Play" Game Design & Lab Analysis
Standard: 7.SP.C.5, 6, 7
Duration: 180 Minutes
The Mogul Mission
Lead the design of a casino game. Calculate the Theoretical Win Chance and run 100 trials to test reality.
PHASE 1 : GAME DESIGN
Game Title & Winning Conditions:
PHASE 2 : THEORETICAL SPECS
Sample Space Map
Theoretical P(Win)
(Show Fraction, Decimal, %)
PHASE 3 : THE 100-TRIAL TEST
PAGE 02 / 04
Experimental Results
Total Wins: _______ / 100
Experimental %: ________
Mogul Analysis Report
"Compare experimental to theory. Why do results differ? Discuss frequency stability across 100 trials."
Student-Facing Rubric
Criteria Expert (4) Practitioner (3) Novice (2/1) Theoretical Specs Sample space is flawless. Probability modeled as fraction, decimal, and % perfectly. Sample space correct; minor conversion errors. Major errors in modeling. Data quality All 100 trials documented. Relative frequency calculated perfectly. 100 trials completed; minor entry slips. Incomplete trials. Report Logic Sophisticated argument citing Law of Large Numbers and statistical variability. Correct comparison; identifies variability accurately. Fails to explain gap. Professionalism Uses correct statistical language and formatting throughout. Neat and complete. Messy or missing parts.
Final Path Score:
/ 16
Teacher Evaluation Key: Path 01
Exemplar Check: Theoretical P(Win)
If student rolls one die for a "Sum of 4 or higher":
Outcomes: {1, 2, 3, 4, 5, 6}
Expected Variability
Students should observe that while their 100 trials may not be exactly 50/50, it should be closer than a sample of only 10 trials would be. This demonstrates 7.SP.C.6 .
Data Detective Project Assignment PATH 02: THE DATA DETECTIVE
Research Case: Uniform Models vs. Reality
Standard: 7.SP.C.7b, 7.SP.C.6
Duration: 180 Minutes
Case Protocol
Prove that real-world outcomes are rarely "uniform." Choose an event (Sports, Weather, or Patterns), record 50 data points , and compare them to a model where all outcomes are equally likely.
STEP 1 : THE RESEARCH SUBJECT
Event Subject & Possible Outcomes:
STEP 2 : THE OBSERVED REALITY
Outcome Tally (Total 50) Rel Freq (%)
DETECTIVE ANALYSIS REPORT
PAGE 02 / 04
The Uniform Model Projection
If all outcomes were Equally Likely , what would the probability be? How many would you expect to see in 50 trials? Show calculation: 1 / [Total Options].
Modeling Proof Area:
The Discrepancy Evidence
Explain the gap between the Uniform Model and Observed Reality . List 3 specific variables causing this bias.
Evidence Factor 01:
Evidence Factor 02:
Evidence Factor 03:
Path 02 Rubric
Criteria Expert (4) Practitioner (3) Novice (2/1) Data Fidelity 50 data points summarized with flawless relative frequency calculations. Most points present; minor conversion slips. Incomplete data set. Uniform Logic Uniform model is perfectly calculated and compared to reality. Model mostly correct; minor slips in counts. Fails to correctly identify uniform chances. Scientific Argument Sophisticated reasoning identifies 3+ variables with clear impact evidence. Logically explains gap; identifying 1-2 factors. Cannot explain gap. Case Report Meticulous, professional, and data-driven throughout.
Tree Architect Project Assignment PATH 03: THE TREE ARCHITECT
Specialization: Compound Event Sample Spaces
Standard: 7.SP.C.8, 7.SP.C.8a/b
Duration: 180 Minutes
Architectural Specs
Visualize the paths of compound probability. Your task: map two complex scenarios using high-precision Tree Diagrams and Outcome Grids .
CASE 1 : THE TRIPLE-STAGE TREE
Design a 3-stage event (e.g., Flip, Spin, Roll). Map every branching outcome clearly below.
Tree Mapping Area
PATH 03 : COORDINATE OUTCOME GRIDS
CASE 2 : THE OUTCOME GRID
"Design a 2-stage event with two distinct tools. Construct an exhaustive grid map."
Construct Grid Matrix Here
Architect's Proof (7.SP.C.8a)
"Identify your 'Target Outcome'. What fraction of the sample space does it occupy?"
Mastery Rubric: Path 03
Criteria Expert (4) Practitioner (3) Novice (2/1) Tree Mapping Diagram flawlessly maps all stages. 100% identification of final outcomes. Diagram correct; minor labeling slips. Incomplete diagram. Grid Accuracy Outcome Table perfectly captures the intersection of two spaces. Grid accurate; 1-2 minor omissions. Fails to correctly organize table. Compound Logic Calculates target probabilities with 100% accuracy using the maps. Correct fraction; minor arithmetic slips. Fails to identify combinations. Precision Crisp, professional, and demo advanced spatial logic. Neat and complete. Messy or unorganized work.
Final Architect Score:
/ 16
Teacher Evaluation Key: Path 03
Case 1 Example: Coin, Die, Choice
A 3x2x2 event should yield 12 branches . The student must list them all as terminal nodes.
Case 2 Example: Die (1-6) and Coin (H-T)
Total outcomes = 12.
Simulator Project Assignment PATH 04: THE SIMULATOR
Specialization: Random Events & Simulation Design
Standard: 7.SP.C.8c, 7.SP.C.6
Duration: 180 Minutes
Simulation Protocol
Simulations allow mathematicians to explore compound events without long-term risk. Design a 3-stage compound event and perform 80 trials to estimate its frequency.
STEP 1 : MODEL ARCHITECTURE
Stage 1 Tool:
Stage 2 Tool:
Stage 3 Tool:
PHASE 2 : THE 80-TRIAL LOG
Simulation Summary
Winning Trials: _______
Sim. Frequency: _______ %
Final Sim Probability Estimate:
________
Reliability Proof
"Analyze simulation accuracy: How would performing 8,000 trials change your frequency results? Reference sample size."
Path 04 Rubric
Criteria Expert (4) Practitioner (3) Novice (2/1) Model Design Appropriate tools perfectly represent a complex 3-stage event. Model logic is flawless. Model correctly represents stages; tool selection is functional. Simulation does not correctly model the event. Execution All 80 trials performed and documented with 100% clarity. Wins highlighted correctly. Trials completed; documentation is mostly clear. Incomplete log or disorganized results. Reasoning Advanced grasp of sample size impact and frequency stability. Logic is sophisticated. Correct understanding of sample size impact. Fails to explain accuracy vs. sample size. Data Fidelity Calculates simulation frequencies flawlessly. Professional units and summaries. Calculations are mostly correct. Fails to correctly summarize data.
Final Simulator Score:
/ 16
Scholar Probability Gauntlet Exam Assignment PATH 05: THE SCHOLAR
Comprehensive Mastery Exam: Probability Gauntlet
Standard: ALL 7.SP.C (MA)
Duration: 180 Minutes
"The Scholar path is a high-rigor gauntlet. You must show exhaustive work, map exhaustive sample spaces, and provide clear algebraic justifications."
I. CHANCE MECHANICS (7.SP.C.5)
Q1. Scaling Logic:
Describe the probability of an event happening as a number between 0 and 1. For each value (0, 1/2, 1), provide a real-world example. Why is 0.5 considered the 'neutral chance' point?
Detailed Response:
The Gauntlet: Part 2
II. PREDICTIVE FREQUENCY (7.SP.C.6)
Q2. District Prediction:
In a random sample of 80 students, 12 were found to be left-handed. Predict the total number of left-handed students in a district of 1,200. Show your algebraic proportion.
Algebraic Proof:
III. MULTI-STAGE SPACES (7.SP.C.8)
Q3. Tree Mapping:
Construct a Tree Diagram for flipping a coin 3 times. What is the probability of getting exactly 2 heads?
Tree Construction Area
The Gauntlet: Part 3
IV. SIMULATION GAUNTLET (7.SP.C.8c)
Q4. Experimental Model Design:
Simulate a "Triple Spinner" (4 colors: R, B, G, Y). Target event: [Red, Blue, Blue] in that order.
Task A: Simulation Tool Mapping
Assign tools (d4, Coin, RNG) to represent the outcomes.
Task B: Theoretical Baseline
Calculate the actual theoretical probability. Show steps.
Path 05 Rubric
Criteria Expert (4) Practitioner (3) Novice (2/1) Spectrum Logic Flawless definitions and mathematically distinct examples for the 0-1 scale. Correct logic; minor inaccuracies in real-world examples. Major errors in probability scale definitions. Predictive Algebra Flawless use of relative frequency to predict population results. All steps shown. Prediction is correct; minor missing steps. Failed to use frequency to make valid prediction. Tree Diagrams Tree Diagram impeccable. 100% identification of the 8 outcomes for 3 flips. Diagram correct; minor labeling error. Diagram is incomplete or flawed. Simulation Modeling Simulation design is robust and demonstrates deep understanding of compound events.
Probability Mastery Teacher Keys Material TEACHER EVALUATION KEYS
Probability Mastery Solutions (MA.7.SP.C)
PATH 01: MOGUL KEY
Theoretical Benchmark
Outcome lists must be exhaustive. If 2 dice: n=36. P(Win) must match correct theoretical sum counts (e.g., Sum=7 is 6/36 or 1/6).
Experimental Check
100 trials should fall within ±5% of theory. Look for discussion of 'frequency stability'—experimental results approach theory as trials increase.
PATH 02: DETECTIVE KEY
Uniform Model Proof
If outcomes (n)=4, Expected Freq = 50 * 0.25 = 12.5 per outcome. Model is P=1/n.
Discrepancy Analysis
Valid factors: "Seasonality for weather," "Skill bias in sports," or "Marketing bias in choices."
PATH 03: ARCHITECT KEY
Case 1 (3x2x2)
Total Terminal Branches = 12. P(Specific Branch) = 1/12 or 8.33%.
Case 2 (d6 + Coin)
Total Cells = 12. P(Heads & Even) = {H2, H4, H6} = 3/12 or 25%.
PATH 04: SIMULATOR KEY
Model Fidelity Benchmark
3-stage quiz (4 choices): P(All Correct) = (1/4)^3 = 1/64 (1.56%). In 80 trials, student should expect ~1.25 winning outcomes. If successes > 10, the model is likely flawed.
PATH 05: SCHOLAR EXAM KEY
Q1 Scale: 0 (Impossible); 0.5 ( neutral chance); 1 (Certain). neutral means equal likelihood for/against.
Q2 Prediction: 12/80 = 0.15. Expected = 1,200 * 0.15 = 180 students.
Q3 Biased: A=100, B=60, C=40. B=45 is within ±25% normal variation; model is robust.
Q4 Map: 8 terminal outcomes (HHH, HHT, HTH, HTT, THH, THT, TTH, TTT). P(2H) = 3/8 (37.5%).
Q5 CHALLENGE KEY
Model: 4 tools (1=R, 2=B, 3=G, 4=Y). Success = Roll sequence [1, 2, 2].
Theo. Prob = (1/4)^3 = 1/64 (~1.56%).
Equation Loop Raider Sheet EQUATION LOOP RAIDER SHEET
Active Learning Protocol / 7.EE.B.4
Raider Name: _________________
Date: _______
Mission Protocol: Solve the equation on your card, show all steps, and look for your answer at the top of another card around the room. If you solve all 20 correctly, you will return to your starting card in a perfect loop!
STATION 01 Card Letter: ______ | Solved Answer: ______
Show Inverse Operations & Checking Step Below
STATION 02 Card Letter: ______ | Solved Answer: ______
Show Inverse Operations & Checking Step Below
STATION 03 Card Letter: ______ | Solved Answer: ______
Show Inverse Operations & Checking Step Below
STATION 04 Card Letter: ______ | Solved Answer: ______
Show Inverse Operations & Checking Step Below
EQUATION LOOP RAIDER // STATIONS 1 - 4
PAGE 01 / 05
The Loop Gauntlet (Part 2)
PAGE 02 / 05
STATION 05 Card Letter: ______ | Solved Answer: ______
STATION 06 Card Letter: ______ | Solved Answer: ______
STATION 07 Card Letter: ______ | Solved Answer: ______
STATION 08 Card Letter: ______ | Solved Answer: ______
EQUATION LOOP RAIDER // STATIONS 5 - 8
PAGE 02 / 05
The Loop Gauntlet (Part 3)
PAGE 03 / 05
STATION 09 Card Letter: ______ | Solved Answer: ______
STATION 10 Card Letter: ______ | Solved Answer: ______
STATION 11 Card Letter: ______ | Solved Answer: ______
STATION 12 Card Letter: ______ | Solved Answer: ______
EQUATION LOOP RAIDER // STATIONS 9 - 12
PAGE 03 / 05
The Loop Gauntlet (Part 4)
PAGE 04 / 05
STATION 13 Card Letter: ______ | Solved Answer: ______
STATION 14 Card Letter: ______ | Solved Answer: ______
STATION 15 Card Letter: ______ | Solved Answer: ______
STATION 16 Card Letter: ______ | Solved Answer: ______
EQUATION LOOP RAIDER // STATIONS 13 - 16
PAGE 04 / 05
The Loop Gauntlet (Part 5)
PAGE 05 / 05
STATION 17 Card Letter: ______ | Solved Answer: ______
STATION 18 Card Letter: ______ | Solved Answer: ______
STATION 19 Card Letter: ______ | Solved Answer: ______
STATION 20 Card Letter: ______ | Solved Answer: ______
Loop Validation: Write your closed loop letter sequence (e.g., A → B → C...): __________________________________ Score: _______ / 20
Equation Loop Raider Cards Station A
Previous Answer Target:
1
Solve for x:
\( 3x - 4.5 = 10.5 \)
Station B
Previous Answer Target:
5
Solve for x:
\( \frac{x}{3} + 1.5 = -1.5 \)
Station C
Previous Answer Target:
-9
Solve for x:
\( -5x + 12 = -3 \)
Station D
Previous Answer Target:
3
Solve for x:
\( 0.5x - 7 = -3 \)
Loop Cards - Page 1 of 5
Station E
Previous Answer Target:
8
Solve for x:
\( \frac{1}{3}x + 8 = 10 \)
Station F
Previous Answer Target:
6
Solve for x:
\( -4x - 6 = 6 \)
Station G
Previous Answer Target:
-3
Solve for x:
\( 1.5x - 8 = 10 \)
Station H
Previous Answer Target:
12
Solve for x:
\( \frac{x}{-2} + 5 = 7 \)
Loop Cards - Page 2 of 5
Station I
Previous Answer Target:
-4
Solve for x:
\( 6x + 2.4 = 14.4 \)
Station J
Previous Answer Target:
2
Solve for x:
\( -2x + 1 = 4 \)
Station K
Previous Answer Target:
-1.5
Solve for x:
\( 0.8x - 3 = 5 \)
Station L
Previous Answer Target:
10
Solve for x:
\( \frac{x}{4} - 1.5 = -2 \)
Loop Cards - Page 3 of 5
Station M
Previous Answer Target:
-2
Solve for x:
\( 4x - 2.5 = 3.5 \)
Station N
Previous Answer Target:
1.5
Solve for x:
\( -3x + 4.5 = 19.5 \)
Station O
Previous Answer Target:
-5
Solve for x:
\( \frac{x}{0.5} + 6 = 14 \)
Station P
Previous Answer Target:
4
Solve for x:
\( 0.2x + 5 = 8 \)
Loop Cards - Page 4 of 5
Equation Loop Raider Key LOOP RAIDER MASTER KEY
Teacher Reference & Setup Manual / 20-Problem Loop
Model Key
Classroom Deployment Guide
1. Print & Cut: Print the 5 pages of loop cards. Cut each page into quarters to get 20 cards.Hang: Tape the cards in a random order around the classroom. Do NOT hang them in alphabetical order, or students won't have to hunt! Solved answers should match the 'Previous Answer Target' of their next card.Start: Hand out the answer sheets. Students can start at any card . If they start at Card M, they write 'M' in the 'Card' box of Station 1, solve it, get 1.5, and look for Card N (which has 1.5 at the top).
The Master Loop Sequence
Use this sequence to instantly verify a student's path. No matter where they start, they must follow this alphabetical order to form a closed loop!
A → B → C → D → E → F → G → H → I → J → K → L → M → N → O → P → Q → R → S → T → A
Detailed Solutions (Stations A - J)
Card A (Starts with 1):
\( 3x - 4.5 = 10.5 \)
Add 4.5: 3x = 15 → x = 5 (Goes to B)
Card B (Starts with 5):
\( \frac{x}{3} + 1.5 = -1.5 \)
Sub 1.5: x/3 = -3 → x = -9 (Goes to C)
Card C (Starts with -9):
\( -5x + 12 = -3 \)
Sub 12: -5x = -15 → x = 3 (Goes to D)
Card D (Starts with 3):
\( 0.5x - 7 = -3 \)
Add 7: 0.5x = 4 → x = 8 (Goes to E)
Card E (Starts with 8):
\( \frac{1}{3}x + 8 = 10 \)
Sub 8: x/3 = 2 → x = 6 (Goes to F)
Card F (Starts with 6):
\( -4x - 6 = 6 \)
Add 6: -4x = 12 → x = -3 (Goes to G)
Card G (Starts with -3):
\( 1.5x - 8 = 10 \)
Add 8: 1.5x = 18 → x = 12 (Goes to H)
Card H (Starts with 12):
\( \frac{x}{-2} + 5 = 7 \)
Sub 5: x/-2 = 2 → x = -4 (Goes to I)
Card I (Starts with -4):
\( 6x + 2.4 = 14.4 \)
Sub 2.4: 6x = 12 → x = 2 (Goes to J)
Card J (Starts with 2):
\( -2x + 1 = 4 \)
Sub 1: -2x = 3 → x = -1.5 (Goes to K)
EQUATION LOOP RAIDER KEY // STATIONS A - J
PAGE 01 / 02
Loop Raider Master Key (Part 2)
PAGE 02 / 02
Detailed Solutions (Stations K - T)
Card K (Starts with -1.5):
\( 0.8x - 3 = 5 \)
Add 3: 0.8x = 8 → x = 10 (Goes to L)
Mastery Project Rubric Project Rubric
20-Point Multi-Dimensional Assessment Scale
Student Name: _________________
Date: _______
Student Protocol: Review the standards below to understand what is required for each point level. Your project will be graded on a 4 (Expert) down to 1 (Novice) scale. Review the criteria before submitting your work to guarantee maximum accuracy!
Category 4 Points (Expert) 3 Points (Practitioner) 2 Points (Apprentice) 1 Point (Novice) Project Completion Fully Complete: All prompts, diagrams, and pages are 100% finished. Zero blank zones.Mostly Complete: 90% of the project is finished. Only 1 or 2 minor questions left blank.Partially Complete: Major sections or several diagrams are left blank. Unfinished components.Incomplete: Only a single page or small section is attempted. Major visuals omitted.Math Calculations Flawless Accuracy: All formulas, math models, and conversions are 100% correct with full steps shown.Minor Slips: Mathematical strategy is correct, but contains 1 or 2 small arithmetic slips.Conceptual Gaps: Multiple formulas are used incorrectly, or work is omitted for key equations.No Proof Shown: Calculations are absent, incorrect, or show zero work/steps.Neatness & Organization Impeccable Order: Handwriting is clean, lines are straight, and work flows logically and beautifully.Neat & Readable: Organized structure. Handwriting is legible and easy for an evaluator to follow.Disorganized: Scrambled layout, hard-to-read segments, or disorganized steps.Messy: Deciphering requires substantial effort. Work is chaotic and scattered.On-Task in Class Fully Engaged: On-task and working productively during 100% of available classroom time.Mostly Engaged: On-task and working independently during the majority of the period.Off-Task Often: Easily distracted. Required multiple verbal redirects from the instructor.Disruptive/Idle: Rarely worked on the project. Disrupted classmates or refused to participate.
