| Curricular Competency |
Explore, analyze, and apply mathematical ideas using reason, technology, and other tools |
Apprenticeship Mathematics 12 |
Reasoning and modelling |
| Keyword: analyze |
Elaboration: examine the structure of and connections between mathematical ideas (e.g., proportional reasoning, metric/imperial conversions) |
| Keyword: reason |
Elaboration: inductive and deductivereasoningpredictions, generalizations, conclusions drawn from experiences (e.g., with puzzles, games, and coding) |
| Keyword: technology |
Elaboration: graphing technology, dynamic geometry, calculators, virtual manipulatives, concept-based appscan be used for a wide variety of purposes, including:exploring and demonstrating mathematical relationshipsorganizing and displaying datagenerating and testing inductive conjecturesmathematical modelling |
| Keyword: other tools |
Elaboration: manipulatives such as rulers and other measuring tools |
|
| Curricular Competency |
Develop thinking strategies to solve puzzles and play games |
Apprenticeship Mathematics 12 |
Reasoning and modelling |
| Keyword: thinking strategies |
Elaboration: using reason to determine winning strategiesgeneralizing and extending |
|
| Big Ideas |
Geometry stories and applications vary across cultures and time. |
Geometry 12 |
No CCG |
| Keyword: Geometry |
Elaboration: 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? |
|
| Big Ideas |
The proving process begins with conjecturing, looking for counter-examples, and refining the conjecture, and the process may end with a written proof. |
Geometry 12 |
No CCG |
| Keyword: proving process |
Elaboration: 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? |
|
| Big Ideas |
Geometry involves creating, testing, and refining definitions. |
Geometry 12 |
No CCG |
| Keyword: definitions |
Elaboration: are seldom the starting point in geometrySample 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? |
|
| Big Ideas |
Finding invariance amidst variation drives geometric investigation. |
Geometry 12 |
No CCG |
| Keyword: invariance amidst variation |
Elaboration: 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? |
|
| Big Ideas |
Diagrams are fundamental to investigating, communicating, and discovering properties and relations in geometry. |
Geometry 12 |
No CCG |
| Keyword: Diagrams |
Elaboration: 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? |
|
| Content |
non-Euclidean geometries |
Geometry 12 |
No CCG |
| Keyword: non-Euclidean geometries |
Elaboration: perspective, spherical, Taxicab, hyperbolictessellations |
|
| Content |
transformations of 2D shapes:- isometries
- non-isometric transformations
|
Geometry 12 |
No CCG |
| Keyword: isometries |
Elaboration: transformations that maintain congruence (translations, rotations, reflections)composition of isometriestessellations |
| Keyword: non-isometric transformations |
Elaboration: dilations and sheartopology |
|
| Content |
constructing tangents |
Geometry 12 |
No CCG |
| Keyword: constructing tangents |
Elaboration: lines tangent to circles, circles tangent to circles, circles tangent to three objects (e.g., points [PPP], three lines [LLL]) |
|
| Content |
circle geometry |
Geometry 12 |
No CCG |
| Keyword: circle geometry |
Elaboration: properties of chords, angles, and tangents to mobilize the proving process |
|
| Content |
parallel and perpendicular lines:- circles as tools in constructions
- perpendicular bisector
|
Geometry 12 |
No CCG |
| Keyword: parallel and perpendicular |
Elaboration: angle bisector |
| Keyword: circles as tools |
Elaboration: constructing equal segments, midpoints |
|
| Content |
geometric constructions |
Geometry 12 |
No CCG |
| Keyword: constructions |
Elaboration: angles, triangles, triangle centres, quadrilaterals |
|
| Curricular Competency |
Incorporate First Peoples worldviews, perspectives, knowledge, and practices to make connections with mathematical concepts |
Geometry 12 |
Connecting and reflecting |
| Keyword: Incorporate |
Elaboration: by:collaborating with Elders and knowledge keepers among local First Peoplesexploring 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 mathematicsexploring cultural practices and knowledge of local First Peoples and identifying mathematical connections |
| Keyword: knowledge |
Elaboration: local knowledge and cultural practices that are appropriate to share and that are non-appropriated |
| Keyword: practices |
Elaboration: 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/) |
|
| Curricular Competency |
Use mistakes as opportunities to advance learning |
Geometry 12 |
Connecting and reflecting |
| Keyword: mistakes |
Elaboration: range from calculation errors to misconceptions |
| Keyword: opportunities to advance learning |
Elaboration: by:analyzing errors to discover misunderstandingsmaking adjustments in further attemptsidentifying not only mistakes but also parts of a solution that are correct |
|