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# Geometry 12

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### Big Ideas

### Grandes idées

Diagrams are fundamental to investigating, communicating, and discovering properties and relations in geometry.

Diagrams

*Sample questions to support inquiry with students:*- How would we describe a specific geometric object to someone who cannot see it?
- What properties can we infer from a diagram?
- What behaviours can we infer from a dynamic diagram?

Finding invariance amidst variation drives geometric investigation.

invariance amidst variation

- Invariance amidst variation can be more easily experienced using current technology and dynamic diagrams. For example, the sum of the angles in planar triangles is invariant no matter what forms a triangle takes.
*Sample questions to support inquiry with students:*- How do we construct geometric shapes that maintain properties under variation?
- What properties change and stay the same when we vary a square, parallelogram, triangle, and so on?
- How can the Pythagorean theorem be restated in terms of variance and invariance?

Geometry involves creating, testing, and refining definitions.

definitions

- are seldom the starting point in geometry
*Sample questions to support inquiry with students:*- How does variation help to refine our definitions of shapes?
- How would we define a square (or a circle) in different ways? When would one definition be better to work with than another?
- How can the definition of a shape be used in constructing the shape?
- How can we modify a definition of a shape to define a new shape?

The proving process begins with conjecturing, looking for counter-examples, and refining the conjecture, and the process may end with a written proof.

proving process

*Sample questions to support inquiry with students:*- Can we make a conjecture about the diagonals of a polygon? Can we find a counter-example to our conjecture?
- How can one conjecture about a
*specific*shape lead to making another more*general*conjecture about a family of shapes? - How can we be sure that a proof is complete?
- Can we find a counter-example to a conjecture?
- How can different proofs bring out different understandings of a relationship?

Geometry stories and applications vary across cultures and time.

Geometry

- Geometry is more than a list of axioms and deductions. Non-Western and modern geometry is concerned with shape and space and is not always axiomatic. It is not always about producing a theorem; rather, it is about modelling mathematical and non-mathematical phenomena using geometric objects and relations. Today geometry is used in a multitude of disciplines, including animation, architecture, biology, carpentry, chemistry, medical imaging, and art.

*Sample questions to support inquiry with students:*- Can we find geometric relationships in local First Peoples art or culture?
- Can we make geometric connections to story, language, or past experiences?
- What do we notice about and how would we construct common shapes found in local First Peoples art?
- How has the notion of “proof” changed over time and in different cultures?
- How are geometric ideas implemented in modern professions?

## Learning Standards

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### Curricular Competencies

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*Students are expected to be able to do the following:*### Reasoning and modelling

Develop thinking strategies to solve puzzles and play games

thinking strategies

- using reason to determine winning strategies
- generalizing and extending

Engage in spatial reasoning in a dynamic environment

spatial reasoning

- being able to think about shapes (real or imagined) and mentally transform them to notice relationships

Explore, analyze, and apply mathematical ideas using reason, technology, and other tools

analyze

- examine the structure of and connections between geometric ideas (e.g., parallel and perpendicular lines, circle geometry, constructing tangents, transformations)

reason

- inductive and deductive reasoning
- predictions, generalizations, conclusions drawn from experiences (e.g., with puzzles, games, and coding)

technology

- graphing technology, dynamic geometry, calculators, virtual manipulatives, concept-based apps
- can be used for a wide variety of purposes, including:
- exploring and demonstrating geometrical relationships
- organizing and displaying data
- generating and testing inductive conjectures
- mathematical modelling

other tools

- paper and scissors, straightedge and compass, ruler, and other concrete materials

Estimate reasonably and demonstrate fluent, flexible, and strategic thinking about number

Estimate reasonably

- be able to defend the reasonableness of an estimated value or a solution to a problem or equation (e.g., congruencies, angles, lengths)

fluent, flexible, and strategic thinking

- being able to generate a family of shapes and apply characteristics across the family

Model with mathematics in situational contexts

Model

- use mathematical concepts and tools to solve problems and make decisions (e.g., in real-life and/or abstract scenarios)
- take a complex, essentially non-mathematical scenario and figure out what mathematical concepts and tools are needed to make sense of it

situational contexts

- including real-life scenarios and open-ended challenges that connect mathematics with everyday life

Think creatively and with curiosity and wonder when exploring problems

Think creatively

- by being open to trying different strategies
- refers to creative and innovative mathematical thinking rather than to representing math in a creative way, such as through art or music

curiosity and wonder

- asking questions to further understanding or to open other avenues of investigation

### Understanding and solving

Develop, demonstrate, and apply conceptual understanding of mathematical ideas through play, story, inquiry, and problem solving

inquiry

- includes structured, guided, and open inquiry
- noticing and wondering
- determining what is needed to make sense of and solve problems

Visualize to explore and illustrate geometric concepts and relationships

Visualize

- create and use mental images to support understanding
- Visualization can be supported using dynamic materials (e.g., graphical relationships and simulations), concrete materials, drawings, and diagrams.

Apply flexible and strategic approaches to solve problems

flexible and strategic approaches

- deciding which mathematical tools to use to solve a problem
- choosing an effective strategy to solve a problem (e.g., guess and check, model, solve a simpler problem, use a chart, use diagrams, role-play)

solve problems

- interpret a situation to identify a problem
- apply mathematics to solve the problem
- analyze and evaluate the solution in terms of the initial context
- repeat this cycle until a solution makes sense

Solve problems with persistence and a positive disposition

persistence and a positive disposition

- not giving up when facing a challenge
- problem solving with vigour and determination

Engage in problem-solving experiences connected with place, story, cultural practices, and perspectives relevant to local First Peoples communities, the local community, and other cultures

connected

- through daily activities, local and traditional practices, popular media and news events, cross-curricular integration
- by posing and solving problems or asking questions about place, stories, and cultural practices

### Communicating and representing

Explain, justify, and evaluate geometric ideas and decisions in many ways

Explain, justify

- use mathematical arguments to convince
- includes anticipating consequences

decisions

- Have students explore which of two scenarios they would choose and then defend their choice.

many ways

- including oral, written, visual, gestures and use of technology
- communicating effectively according to what is being communicated and to whom

Represent mathematical ideas in concrete, pictorial, and symbolic forms

Represent

- concretely, diagrammatically, symbolically, including using models, tables, graphs, words, numbers, symbols

Use geometric vocabulary and language to contribute to discussions in the classroom

discussions

- partner talks, small-group discussions, teacher-student conferences

Take risks when offering ideas in classroom discourse

discourse

- is valuable for deepening understanding of concepts
- can help clarify students’ thinking, even if they are not sure about an idea or have misconceptions

### Connecting and reflecting

Reflect on geometric thinking

Reflect

- share the geometric thinking of self and others, including evaluating strategies and solutions, finding counter-examples, extending, posing new problems and questions, proving results

Connect mathematical concepts with each other, other areas, and personal interests

Connect mathematical concepts

- to develop a sense of how mathematics helps us understand ourselves and the world around us (e.g., daily activities, local and traditional practices, popular media and news events, social justice, cross-curricular integration)

Use mistakes as opportunities to advance learning

mistakes

- range from calculation errors to misconceptions

opportunities to advance learning

- by:
- analyzing errors to discover misunderstandings
- making adjustments in further attempts
- identifying not only mistakes but also parts of a solution that are correct

Incorporate First Peoples worldviews, perspectives, knowledge, and practices to make connections with mathematical concepts

Incorporate

- by:
- collaborating with Elders and knowledge keepers among local First Peoples
- exploring the First Peoples Principles of Learning (http://www.fnesc.ca/wp/wp-content/uploads/2015/09/PUB-LFP-POSTER-Princip... e.g., Learning is holistic, reflexive, reflective, experiential, and relational [focused on connectedness, on reciprocal relationships, and a sense of place]; Learning involves patience and time)
- making explicit connections with learning mathematics
- exploring cultural practices and knowledge of local First Peoples and identifying mathematical connections

knowledge

- local knowledge and cultural practices that are appropriate to share and that are non-appropriated

practices

- Bishop’s cultural practices: counting, measuring, locating, designing, playing, explaining (http://www.csus.edu/indiv/o/oreyd/ACP.htm_files/abishop.htm)
- Aboriginal Education Resources (www.aboriginaleducation.ca)
*Teaching Mathematics in a First Nations Context*, FNESC (http://www.fnesc.ca/resources/math-first-peoples/)

### Content

*Students are expected to know the following:*geometric constructions

constructions

- angles, triangles, triangle centres, quadrilaterals

parallel and perpendicular lines:

parallel and perpendicular

- angle bisector

- circles as toolsin constructionscircles as tools
- constructing equal segments, midpoints

- perpendicular bisector

circle geometry

circle geometry

- properties of chords, angles, and tangents to mobilize the proving process

constructing tangents

constructing tangents

- lines tangent to circles, circles tangent to circles, circles tangent to three objects (e.g., points [PPP], three lines [LLL])

transformations of 2D shapes:

- isometriesisometries
- transformations that maintain congruence (translations, rotations, reflections)
- composition of isometries
- tessellations

- non-isometric transformationsnon-isometric transformations
- dilations and shear
- topology

non-Euclidean geometries

non-Euclidean geometries

- perspective, spherical, Taxicab, hyperbolic
- tessellations

**Note:**Some of the learning standards in the PHE curriculum address topics that some students and their parents or guardians may feel more comfortable addressing at home. Refer to ministry policy regarding opting for alternative delivery.