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# Foundations of Mathematics 12

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### Big Ideas

### Grandes idées

Probabilistic thinking informs decision making in situations involving chance and uncertainty.

Probabilistic thinking

*Sample questions to support inquiry with students:*- How do we make decisions involving probabilities?
- How reliable is a test that is 98% accurate?
- What is the difference between reliability and accuracy?
- What information is needed when considering the likelihood of an event?

Modelling data requires an understanding of a variety of functions.

Modelling

*Sample questions to support inquiry with students:*- How do we know what type of regression best models a given set of data?
- What factors would affect the reliability of a regression analysis?
- What are the limitations associated with regression models?

Mathematical analysis informs financial decisions.

decisions

*Sample questions to support inquiry with students:*- How do we make decisions regarding our financial options?
- What are the repercussions of our financial decisions (e.g., in the short term versus the long term)?
- What factors influence our willingness to take financial risks?

Through explorations of spatial relationships, we can develop a geometrical appreciation of the world around us.

explorations

*Sample questions to support inquiry with students:*- What can we construct using a straightedge and compass?
- What properties change and stay the same when we vary a square, parallelogram, triangle, and so on?
- How are circles, ellipses, parabolas, and hyperbolas related?
- Where are conics found in the world around us?
- How does nature exhibit fractal properties?
- What patterns do we see in fractals?

## 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

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

analyze

- examine the structure of and connections between mathematical ideas (e.g., conic sections, functions, financial planning)

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 mathematical relationships
- organizing and displaying data
- generating and testing inductive conjectures
- mathematical modelling

other tools

- manipulatives such as algebra tiles 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., regression analysis and combinatorics calculations)

fluent, flexible, and strategic thinking

- includes using known facts and benchmarks; partitioning; applying whole number strategies to graphing; regression choice; probability

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 context

- 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 mathematical 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 appropriate 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 and justify mathematical ideas and decisions in many ways

Explain and 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, use of technology
- communicating effectively according to what is being communicated and to whom

Represent mathematical ideas in concrete, pictorial, and symbolic forms

Represent

- using models, tables, graphs, words, numbers, symbols
- connecting meanings among various representations

Use mathematical 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 mathematical thinking

Reflect

- share the mathematical thinking of self and others, including evaluating strategies and solutions, extending, posing new problems and questions

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 explorations:

- constructionsconstructions
- perpendicular bisector, tangents, polygons, tessellations, geometric art

- conicsconics
- locus definition and constructions, conic sections, applications

- fractalsfractals
- understanding fractals as an iteration of a simple instruction
- constructing and analyzing models of fractals, such as Cantor’s dust, Serpinski’s triangle, Koch’s snowflake
- connecting fractals with nature

graphical representations of polynomial, logarithmic, exponential, and sinusoidal functions

representations

- using technology only
- using characteristics of a graph to identify these functions

regression analysis

regression analysis

- polynomial, exponential, sinusoidal, logarithmic
- applying the appropriate regression model

combinatorics

combinatorics

- permutations, combinations, pathways, Pascal’s Triangle

odds, probability, and expected value

odds, probability

- mutually exclusive, non–mutually exclusive, conditional probability, binomial probability
- Venn diagrams

financial planning

financial planning

- developing a personal financial portfolio
- mortgages
- risk
- changing interest rates and/or payments
- credit cards
- exploring banking options and financial markets

**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.