Course Scope and Sequence: Geometry Summer Program

Overview: This 12-day (6 hours/day, 4 days/week, 3 weeks) course targets high school students who failed geometry, focusing on hands-on applications in woodshop and clay to engage disengaged learners. Weeks 2 and 3 include a 4-week gap for video-based design work.

Week 1: Foundations of Geometric Design (June 15–18)

Day 1: 2D Geometry in Design & Basic Elements

• Content: Course introduction, woodshop safety. Review points, lines, planes, segments, rays. Types of angles (acute, obtuse, right, straight, reflex) and relationships (complementary, supplementary, vertical). Use calipers, protractors for measurement. Design activity: Compare isometric, orthographic, perspective drawings of 3D objects. Design a slanted shelf or clay form with specific angles. Intro to CAD for project ideation.
• Formulas:
  • Complementary angles: \(\angle 1 + \angle 2 = 90^\circ\)
  • Supplementary angles: \(\angle 1 + \angle 2 = 180^\circ\)
  • Vertical angles: \(\angle 1 = \angle 2\)
• Vocabulary: Point, line, plane, segment, ray, angle (acute, obtuse, right, straight, reflex), complementary, supplementary, vertical, isometric, orthographic, perspective, caliper, protractor.
• Time:
  • 1 hr: Intro, safety, basic elements lecture.
  • 1.5 hrs: Angle types/relationships, measurement tools practice.
  • 1.5 hrs: Design activity (drawings, shelf/form design).
  • 2 hrs: CAD intro, project ideation.

Day 2: 2D Polygons: Triangles, Quadrilaterals, Circles

• Content: Properties of triangles (scalene, isosceles, equilateral, right), Triangle Inequality Theorem. Quadrilaterals (parallelogram, rectangle, square, rhombus, trapezoid). Circles: radius, diameter, circumference, area, arcs, chords. Design activity: Create a collage of polygonal/circular objects (e.g., triangular prism shelf, rectangular box) using CAD. Clay slab-building to scale, emphasizing measurement.
• Formulas:
  • Triangle area: \(A = \frac{1}{2}bh\)
  • Rectangle area: \(A = lw\)
  • Square area: \(A = s^2\)
  • Parallelogram area: \(A = bh\)
  • Trapezoid area: \(A = \frac{1}{2}(b_1 + b_2)h\)
  • Circle circumference: \(C = 2\pi r\), \(C = \pi d\)
  • Circle area: \(A = \pi r^2\)
• Vocabulary: Scalene, isosceles, equilateral, right triangle, Triangle Inequality Theorem, parallelogram, rectangle, square, rhombus, trapezoid, radius, diameter, circumference, area, arc, chord, tangent, secant.
• Time:
  • 1.5 hrs: Lecture/practice on triangles, quadrilaterals, circles.
  • 2 hrs: Design activity (CAD collage, dimensioning).
  • 2.5 hrs: Clay slab-building, measurement.

Day 3: 3D Spheres, Cylinders, Cones

• Content: Properties of spheres, cylinders, cones. Derive formulas for surface area and volume. Use protractors, calipers for precision. Design activity: Create orthographic drawings of a clay vessel. Application: Wood lathe (safety, segment cylinder to specs). Potter’s wheel (calculate volume/surface area of clay sphere, cone up, open into cylinder, measure wall thickness).
• Formulas:
  • Cylinder volume: \(V = \pi r^2 h\)
  • Cylinder surface area: \(SA = 2\pi r h + 2\pi r^2\)
  • Sphere volume: \(V = \frac{4}{3}\pi r^3\)
  • Sphere surface area: \(SA = 4\pi r^2\)
  • Cone volume: \(V = \frac{1}{3}\pi r^2 h\)
  • Cone surface area: \(SA = \pi r l + \pi r^2\) (l = slant height)
• Vocabulary: Sphere, cylinder, cone, volume, surface area, lateral area, slant height, orthographic, lathe, potter’s wheel, caliper.
• Time:
  • 1.5 hrs: Lecture on 3D shapes, formula derivation.
  • 1.5 hrs: Design activity (vessel drawings).
  • 3 hrs: Lab (lathe: 1.5 hrs, potter’s wheel: 1.5 hrs).

Day 4: 3D Solid Figures: Prisms and Pyramids

• Content: Properties of prisms (rectangular, triangular) and pyramids. Derive surface area/volume formulas. Design activity: CAD design of prismatic/pyramidal objects (e.g., wooden box with triangular lid, clay pyramid). Building: Construct designs in wood/clay. Assessment: Short quiz on 2D/3D formulas and angle relationships.
• Formulas:
  • Rectangular prism volume: \(V = lwh\)
  • Rectangular prism surface area: \(SA = 2(lw + lh + wh)\)
  • Triangular prism volume: \(V = \frac{1}{2}bh \cdot H\) (H = prism height)
  • Triangular prism surface area: Sum of face areas
  • Pyramid volume: \(V = \frac{1}{3}Bh\) (B = base area)
  • Pyramid surface area: Sum of base and lateral face areas
• Vocabulary: Prism, pyramid, rectangular prism, triangular prism, base, height, surface area, volume, CAD.
• Time:
  • 1 hr: Lecture on prisms/pyramids.
  • 1.5 hrs: CAD design activity.
  • 2.5 hrs: Building in wood/clay.
  • 1 hr: Quiz (formulas, angle relationships).
• Assessment: Quiz (20 questions, multiple-choice/short-answer) on days 1–3 formulas and concepts.

Week 2: Applying Geometry in Three Dimensions (August 11–14)

Day 5: Review 2D Geometry

• Content: Review points, lines, angles, triangles, quadrilaterals, circles. Reinforce measurement with calipers/protractors. Design activity: Revise week 1 designs (e.g., shelf, collage) based on video homework from gap. Building: Begin small-scale 2D-based objects (e.g., wooden frames, clay tiles) to refresh skills.
• Formulas (Reviewed):
  • Angle relationships: \(\angle 1 + \angle 2 = 90^\circ\), \(\angle 1 + \angle 2 = 180^\circ\), \(\angle 1 = \angle 2\)
  • Triangle area: \(A = \frac{1}{2}bh\)
  • Quadrilateral areas: \(A = lw\), \(A = s^2\), \(A = bh\), \(A = \frac{1}{2}(b_1 + b_2)h\)
  • Circle: \(C = 2\pi r\), \(A = \pi r^2\)
• Vocabulary: Point, line, angle, triangle, quadrilateral, circle, complementary, supplementary, vertical, perimeter, area, radius, diameter.
• Time:
  • 1.5 hrs: Review lecture, measurement practice.
  • 2 hrs: Design revision in CAD.
  • 2.5 hrs: Building (frames, tiles).

Day 6: Review 3D Shapes: Cylinders, Cones, Spheres

• Content: Review properties and formulas for cylinders, cones, spheres. Practice calculations with real objects (e.g., measure clay spheres). Design activity: Refine week 1 vessel designs, incorporating feedback. Building: Continue lathe/wheel work, focusing on precision (e.g., consistent cylinder dimensions).
• Formulas (Reviewed):
  • Cylinder: \(V = \pi r^2 h\), \(SA = 2\pi r h + 2\pi r^2\)
  • Sphere: \(V = \frac{4}{3}\pi r^3\), \(SA = 4\pi r^2\)
  • Cone: \(V = \frac{1}{3}\pi r^2 h\), \(SA = \pi r l + \pi r^2\)
• Vocabulary: Cylinder, cone, sphere, volume, surface area, slant height, precision, dimension.
• Time:
  • 1.5 hrs: Review lecture, measurement practice.
  • 2 hrs: Design refinement in CAD.
  • 2.5 hrs: Building (lathe, wheel).

Day 7: Transformations, Congruence, Symmetry

• Content: Introduce translations, rotations, reflections, line/rotational symmetry. Congruent figures (equal corresponding sides/angles). Design activity: Create tessellated wooden patterns or clay pieces with symmetry (e.g., hexagonal tiles). Building: Construct symmetrical designs, ensuring precision.
• Formulas:
  • None new; apply prior formulas (e.g., \(A = \frac{1}{2}bh\)) to verify congruence.
• Vocabulary: Translation, rotation, reflection, line symmetry, rotational symmetry, congruence, tessellation, corresponding angles/sides.
• Time:
  • 1.5 hrs: Lecture on transformations, symmetry, congruence.
  • 2 hrs: Design activity (tessellations in CAD).
  • 2.5 hrs: Building (wood/clay).

Day 8: Congruence and Similarity

• Content: Review congruence. Introduce similarity (proportional sides, equal angles) and scale factors. Design activity: Create scaled versions of objects (e.g., small/large boxes). Building: Construct similar objects, ensuring proportional measurements. Assessment: Project checkpoint (present designs, explain congruence/similarity).
• Formulas:
  • Scale factor for similar figures: Ratio of corresponding side lengths
  • Area ratio for similar figures: \((\text{scale factor})^2\)
• Vocabulary: Similarity, scale factor, proportional, corresponding sides/angles.
• Time:
  • 1.5 hrs: Lecture on similarity, scale factors.
  • 1.5 hrs: Design activity (scaled objects in CAD).
  • 2 hrs: Building.
  • 1 hr: Assessment (present designs).
• Assessment: Students present CAD designs, explaining congruence/similarity applications.

Week 3: Advanced Applications and Project Completion (August 18–21)

Day 9: Coordinate Geometry and Design

• Content: Introduce coordinate plane, plotting points, defining shapes with coordinates. Distance and midpoint formulas. Design activity: Create a complex object (e.g., modular shelf) using coordinates in CAD. Building: Begin constructing coordinate-based designs. Supplemental lesson: Intro to right triangle trigonometry (sine, cosine, tangent).
• Formulas:
  • Distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
  • Midpoint formula: \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\)
  • Trigonometric ratios: \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\)
• Vocabulary: Coordinate plane, x-axis, y-axis, ordered pair, distance, midpoint, sine, cosine, tangent, hypotenuse.
• Time:
  • 1 hr: Lecture on coordinate geometry.
  • 0.5 hr: Supplemental lesson (trigonometry intro).
  • 2 hrs: Design activity (CAD with coordinates).
  • 2.5 hrs: Building.

Day 10: Geometric Constraints and Optimization

• Content: Discuss constraints (parallel/perpendicular lines, material limits) and optimization (minimize material, maximize strength). Design activity: Refine projects based on constraints (e.g., adjust shelf angles for stability). Building: Continue project construction. Supplemental lesson: Circle theorems (e.g., inscribed angles, central angles).
• Formulas:
  • Inscribed angle theorem: \(\text{Inscribed angle} = \frac{1}{2} \times \text{intercepted arc}\)
  • Central angle: \(\text{Central angle} = \text{measure of arc}\)
• Vocabulary: Constraint, parallel, perpendicular, optimization, material efficiency, inscribed angle, central angle, arc measure.
• Time:
  • 1 hr: Lecture on constraints/optimization.
  • 0.5 hr: Supplemental lesson (circle theorems).
  • 2 hrs: Design refinement in CAD.
  • 2.5 hrs: Building.

Day 11: Final Project Work and Refinement

• Content: Dedicated building time for project completion. Teacher supports problem-solving (e.g., fixing misaligned joints). Supplemental lesson: Pythagorean theorem and its applications in design checks. Final prep for presentations.
• Formulas:
  • Pythagorean theorem: \(a^2 + b^2 = c^2\)
• Vocabulary: Pythagorean theorem, right triangle, leg, hypotenuse, alignment.
• Time:
  • 0.5 hr: Supplemental lesson (Pythagorean theorem).
  • 4.5 hrs: Building, teacher support.
  • 1 hr: Presentation prep (outline geometric principles used).

Day 12: Project Completion, Presentation, and Reflection

• Content: Final project touches. Students present objects, explaining geometric principles (e.g., formulas, symmetry) used in design/construction. Course reflection and feedback. Assessment: Presentation and project evaluation.
• Formulas: None new; students reference prior formulas in presentations.
• Vocabulary: Presentation, geometric application, reflection.
• Time:
  • 1 hr: Final touches.
  • 4 hrs: Presentations (15 min/student, assuming ~16 students).
  • 1 hr: Reflection, feedback.
• Assessment: Evaluate projects/presentations on geometric accuracy, creativity/quality, and clarity in explaining principles.