Big Ideas
Big Ideas
Diagrams are fundamental to investigating, communicating, and discovering properties and relations in geometry.
- 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 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.
- 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.
- 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 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?
Content
Learning Standards
Content
geometric constructions
- angles, triangles, triangle centres, quadrilaterals
parallel and perpendicular lines:
- angle bisector
- circles as toolsin constructions
- constructing equal segments, midpoints
- perpendicular bisector
circle geometry
- properties of chords, angles, and tangents to mobilize the proving process
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:
- isometries
- transformations that maintain congruence (translations, rotations, reflections)
- composition of isometries
- tessellations
- non-isometric transformations
- dilations and shear
- topology
non-Euclidean geometries
- perspective, spherical, Taxicab, hyperbolic
- tessellations
Curricular Competency
Learning Standards
Curricular Competency
Reasoning and modelling
Develop thinking strategies to solve puzzles and play games
- using reason to determine winning strategies
- generalizing and extending
Engage in spatial reasoning in a dynamic environment
- 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
- examine the structure of and connections between geometric ideas (e.g., parallel and perpendicular lines, circle geometry, constructing tangents, transformations)
- inductive and deductive reasoning
- predictions, generalizations, conclusions drawn from experiences (e.g., with puzzles, games, and coding)
- 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
- paper and scissors, straightedge and compass, ruler, and other concrete materials
Estimate reasonably and demonstrate fluent, flexible, and strategic thinking about number
- be able to defend the reasonableness of an estimated value or a solution to a problem or equation (e.g., congruencies, angles, lengths)
- being able to generate a family of shapes and apply characteristics across the family
Model with mathematics in situational contexts
- 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
- including real-life scenarios and open-ended challenges that connect mathematics with everyday life
Think creatively and with curiosity and wonder when exploring problems
- 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
- 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
- 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
- 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
- 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)
- 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
- 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
- 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
- use mathematical arguments to convince
- includes anticipating consequences
- Have students explore which of two scenarios they would choose and then defend their choice.
- 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
- concretely, diagrammatically, symbolically, including using models, tables, graphs, words, numbers, symbols
Use geometric vocabulary and language to contribute to discussions in the classroom
- partner talks, small-group discussions, teacher-student conferences
Take risks when offering ideas in classroom 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
- 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
- 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
- range from calculation errors to misconceptions
- 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
- 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-Princi…; 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
- local knowledge and cultural practices that are appropriate to share and that are non-appropriated
- 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/)