342A Unit planning + 3 lessons

 EDCP 342A Unit planning: Rationale and overview for planning a unit of work in secondary school mathematics

Your name: Yikang Zong
School, grade & course: Richmond Secondary School, IB Math 11 AA SL

Topic of unit (NOTE: This should be a unit you will actually be teaching on practicum!):

11. Relationships in space: Geometry and trigonometry in 2D and 3D

 & 12.1 Radian measure, arcs, sectors and segments

Preplanning questions:

(1) Why do we teach this unit to secondary school students? Research and talk about the following: Why is this topic included in the curriculum? Why is it important that students learn it? What learning do you hope they will take with them from this? What is intrinsically interesting, useful, beautiful about this topic? (150 words) This unit is included in the secondary curriculum to help students develop spatial reasoning, a key skill linked in research to success in STEM fields such as engineering, architecture, computer graphics, and physics.  Trigonometry is widely applied in Physics 11 & 12, related to force analysis and other topics, it is also a large portion of functions and graphs, as well as calculus. Later, when students get into STEM in post-secondary education, they also appear in many fields and are critically applied in almost all engineering courses.  After this segment, students should have a thorough understanding of how to describe 2D and 3D shapes in coordinate system, understand the correct terminologies, familiar with applying trigonometry properties and laws to solve geometry analyse problems. Trigonometry is intrinsically beautiful in its rigid relations, the methods used to develop it are great ways to train logical deduction/induction skills. They are extremely useful when applied to dissect and analyse geometric shapes (i.e., tessellation), also their wave-like function properties are widely applied in describing wave-like phenomena.

(2) A mathematics project connected to this unit: Plan and describe a student mathematics project that will form part of this unit. Describe the topic, aims, process and timing, and what the students will be asked to produce, and how you will assess the project. (250 words) I plan to have students build an origami structure of a polygon or polyhedron using triangular tessellation. They can use computer modelling programs or AI to aid their construction, but the final product must be a real origami construct. The aim is to teach students how they can dissect almost all 2D and 3D shapes into triangular shapes, and through analysing the triangular properties, all these shapes can be accurately described and reduced to triangles. This also teaches them basic logic in current computer graphics algorithms, if they are ever interested in CG applications in either art, modelling, or image detection & analysis fields, this project will give them a head start to familiarize themselves with their future projects or research. The project will be introduced after the first week of lectures, and the deadline is set in 3 weeks, submitted in a short written analysis and paper construct. After the first week’s lesson on 2D and 3D shapes, they should have some understanding of the correct terminology and basic properties to start their research. The 3 weekends should be sufficient for them to start their research, then plan their end product and methods, and finally build their structure. The project will be assessed based on its written analysis and paper construct. The written analysis should include their research process and how they planned to build the structure. The paper's construct, if in 2D should show clear tessellation, if in 3D should demonstrate its structural integrity, and explain its relation with the triangular shape’s rigidity.

(3) Assessment and evaluation: How will you build a fair and well-rounded assessment and evaluation plan for this unit? Include formative and summative, informal/ observational and more formal assessment modes. (100 words) Realistically, all assessments we do in IB outside of IA and EA are formative. This segment will be assessed in homeworks, project, and standardized written tests initially. If they do not perform well in these criteria, I plan to provide an optional oral assessment in addition to give students a chance to show their understanding by observing their logical build-up.  The timing of this unit is also suitable for them to come up with a draft for their IA paper. If they wish to start the process, an initial IA draft can be submitted and marked to take over the weight of the project or oral assessment, and I will review and provide feedback on this draft. IA is a critical portion of their IB course, I think this alternative can help lighten their already intense timetable and provide more incremental build-up for their summative assessment.

Elements of your unit plan:

a)  Give a numbered list of the topics of the 10-12 lessons in this unit in the order you would teach them. 

Lesson	Topic
1	Geometry of 3D Shapes
2	Right-angled triangle trigonometry
3	Modelling and investigation, Project introduction, Q&A
4	Sine rule and applications
5	Cosine rule and histories
6	Unit quiz
7	Application of right and non-right-angled trigonometry
8	Quiz feedback and chapter review
9	Unit test
10	Project feedback/ Oral assessment/ IA draft feedback
(11)	Radian measure, arcs, sectors, and segments
(12)	Application of Radian measure, practices

b) Write a detailed lesson plan for three of the lessons which will not be in a traditional lecture/ exercise/ homework format.  These three lessons should include at least three of the following six elements related to your mathematical topic. (And of course, you could include more than three!) 

These elements should be thoroughly integrated into the lessons (i.e. not an add-on that the teacher just tells!)

a) History of this mathematics

b) Arts and mathematics

c) Indigenous perspectives and cultures

d) Social/environmental justice 

e) Open-ended problem solving in groups at vertical erasable surfaces (“thinking classroom”)

f) Telling only what is arbitrary, and having students work on what is logically ‘necessary’

Be sure to include your pedagogical goals, topic of the lesson, preparation and materials, approximate timings, an account of what the students and teacher will be doing throughout the lesson, and ways that you will assess students’ background knowledge, student learning and the overall effectiveness of the lesson. Please use a template that you find helpful, and that includes all these elements.

Modelling and investigation 3D Forms Through Sculpture and Shadow

Integrates: (b) Arts & mathematics, (e) Open-ended problem solving, (f) Arbitrary vs. necessary
Time: 80 minutes
Topic: 3D geometry (Construction, volumes, angles between edges and faces)

Pedagogical Goal: Students develop their understanding of 3D shapes connection to 2D shapes by plane intersection

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Preparation & Materials

• Paper for sculpting, scissors, tape
  
  

• Flashlights to create shadows
  
  

• 3D geometric solids (physical models)
  
  

• Vertical whiteboards
  
  

• Photographs of Indigenous carving, Islamic tiling, and contemporary sculpture showing geometric forms
  
  

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Lesson Flow

1. Visual Spark (10 min)

The teacher shows sculptures and carvings with strong geometric foundations.
Students identify shapes, angles, and edges.

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2. Art-Building Challenge (15 min)

Groups are given the challenge:

“Create a 3D sculpture whose shadow forms a perfect equilateral triangle at a certain angle, it cannot be a flat piece of paper cut into an equilateral triangle.”

No formulas given.
Students choose materials and begin experimenting with fixtures, folding, and angles.

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3. Vertical Surfaces — Geometry Reasoning (25 min)

Groups rotate to boards and explain:

• How they determined which shapes might cast the desired shadow
  
  

• What geometric properties matter
  
  

• What is arbitrary (material, colour, orientation)
  
  

• What is necessary (angle measures, edge lengths, face relationships)
  
  

Teacher prompts thinking:

• “How can you find the height?”
  
  

• “How do you know your faces are congruent?”
  
  

• “What angles must be preserved for the shadow to be equilateral?”
  
  

This forces students to reason about angles between lines/planes.

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4. Testing & Refinement (20 min)

Students shine flashlights on their sculpture to test the shadow.
They adjust geometry based on feedback.

This builds:

• Spatial reasoning
  
  

• Trigonometric thinking about projection
  
  

• Informal understanding of 3D relationships
  
  

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5. Introduce Project, Q&A (10 min)

• Project: Build an origami structure of a polygon or polyhedron using triangular tessellation. They can use computer modelling programs or AI to aid their construction, but the final product must be a real origami construct.

• The project will be assessed based on its written analysis and paper construct. The written analysis should include their research process and how they planned to build the structure. The paper's construct, if in 2D should show clear tessellation, if in 3D should demonstrate its structural integrity, and explain its relation with the triangular shape’s rigidity.

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Assessment

• Teacher observations of problem-solving
  
  

• Shadow accuracy
  
  

• Student explanations using angle vocabulary
  
  

• Creativity + correctness in final sculptures

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Cosine Rule and Histories

Integrates: (a) History of this mathematics, (e) Open-ended problem solving, (f) Arbitrary vs. necessary
Time: 80 minutes
Topic: Cosine rule

Pedagogical Goal: Students develop their skills in formulating logic proofs, scaffolded with similar example

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Preparation & Materials

• Lecture slide with activation problem, proof, and includes Al-Kashi's Theorem
  
  

• Vertical whiteboards and markers
  
  
  

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Lesson Flow

1. Activation Problem (15 min)

First slide shows a geometric problem from Waterloo’s Weekly Problem. 

(current for example: https://cemc.uwaterloo.ca/sites/default/files/documents/2025/POTWE-25-G-11-P.html)

• Students have an attempt at solving it, the answer will be provided in the following slide.
  
  

• Gear up their logical thinking

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2. Step-by-step Proof of Cosine Rule, and history of Al-Kashi’s Theorem (25 min)

Handwritten and explained proof of the Cosine Rule for acute triangles, and slides on Persian mathematician Jamshīd al-Kāshī’s work for triangulation, leading to its name "Al-Kashi's Theorem" in some parts of the world.

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3. Vertical Surfaces — Student proof the rule for obtuse triangles (20 min)

Groups rotate to boards, discuss and show proof:

• How they approach an obtuse triangle
  
  

• What geometric properties matter
  
  

• What is arbitrary, which side/angles to start with
  
  

• What is necessary, how to dissect/scaffold to prove for the obtuse triangle
  
  

Teacher prompts thinking:

• “How do you start the approach?”
  
  

• “How do you dissect your triangle?”
  
  

• “What angles and lines to use in intermediate steps?”
  
  

This forces students to reason their tessellation of a non-right triangle shape.

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4. Gallery walk and feedback  (20 min)

Students explore other groups' approaches.
They adjust their proof based on feedback.

This builds:

• Communicative through math proof
  
  

• Trigonometric thinking about tessellation
  
  

• Understanding of applying the right-triangle rules to non-right-triangles
  
  

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Assessment

• Teacher observations of problem-solving
  
  

• Approaches to proof, and correctness
  
  

• Student explanations on their approach
  
  

Radian Measure

Integrates: (a) History of this mathematics, (b) Art and mathematics, (e) Open-ended problem solving
Time: 80 minutes
Topic: Radian measure

Pedagogical Goal: Students learn the importance and application of radian and apply it in art creation 

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Preparation & Materials

• Lecture slides with an activation problem, definition of radians, and include Thomas Muir’s first use of radian. 

• Video shows the animated circle’s radian-to-degree connection
  
  

• Papers and markers for origami
  
  
  

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Lesson Flow

1. Activation Problem (15 min)

First slide shows some “which one doesn’t belong” diagrams from the WODB website’s shapes section. 

(current, for example: https://talkingmathwithkids.com/wodb-shapes/)

• Students give their choice and reasons, no universally correct answer, of course.
  
  

• Shows there are multiple ways to analyse shapes.

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2. Definition of Radian, and history of Radian  (35 min)

Explains the definition of Radian measures, visualised with video animation, and slides on James Thomson and Thomas Muir, with its use gaining widespread acceptance over time, particularly with the rise of calculus in the 17th century and its formalization in the 19th century. 

• Show that Radian is an alternative way to analyse circles, connecting open-mindedness in problem-solving

• Early introduction to the radian’s advancement in calculus applications, explaining its necessity

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3. Art creation - students create origami art (25 min)

Students learn to fold origami art, focusing on dissecting circular arcs.

Tutorial used: https://origami.me/fortune-teller/

• Students learn circular dissection in art pieces
  
  

• Connect the radian concept with hands-on experience
  
  

• Shows alternate problem-solving techniques
  
  

• Learn the concept of circle dissection that existed long in historical art pieces 

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4. Exit-slip  (5 min)

Students write their understanding of the Radian definition and why they are useful

This reinforces:

• Their learning by re-expressing the concept
  
  

• Their understanding of radian’s applications
  
  
  

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Assessment

• Teacher observations of the ability to identify differences
  
  

• Teacher observations of learning a new concept through applications
  
  

• Students’ communicative skills through exit slips