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# Foundations of Mathematics 11

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### Big Ideas

### Grandes idées

Similar shapes and objects have proportional relationships that can be described, measured, and compared.

Similar

*Sample questions to support inquiry with students:*- What characteristics make objects similar?
- How do the properties of 3D objects change in an enlargement or a reduction?
- How do the properties of 2D objects change in an enlargement or a reduction?

Optimization informs the decision-making process in situations involving extreme values.

Optimization

- a mathematical analysis used to determine the minimum or maximum output for a given situation
*Sample**questions to support inquiry with students:*- Can we think of a story where a conflict can be resolved through optimization?
- How can mathematics help us make decisions regarding the best course of action?
- What factors influence the decision-making process when determining an optimal solution?
- How do graphs aid in understanding a situation that is being optimized?

Logical reasoning helps us discover and describe mathematical truths.

Logical reasoning

- the process of using a strategic, systematic series of steps based on valid mathematical procedures and given statements to form a conclusion
*Sample**questions to support inquiry with students:*- How can logical reasoning help us deal with problems in our everyday lives?
- How does puzzle and game analysis help us in the world outside the math classroom?

Statistical analysis allows us to notice, wonder about, and answer questions about variation.

variation

- occurs in observation (e.g., reaction to medications, opinions on topics, income levels, graduation rates)
*Sample**questions to support inquiry with students:*- How do we gather data in order to answer questions?
- How do we analyze data and make decisions?
- Can we think of a story that involves variation? How would we describe the variation?
- When analyzing data, what are some of the factors that need to be considered before making inferences?

## Learning Standards

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### Curricular Competencies

*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., quadratics and cubic functions, linear inequalities, optimization, financial decision making)

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., angle size reasonableness, scale calculations and unit choice, optimal solutions)

fluent, flexible, and strategic thinking

- includes:
- using known facts and benchmarks, partitioning, applying whole number strategies to rational numbers and algebraic expressions
- choosing from different ways to think of a number or operation (e.g., Which will be the most strategic or efficient?)

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 mathematical understanding 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:*forms of mathematical reasoning

mathematical reasoning

- logic, conjecturing, inductive and deductive thinking, proofs, game/puzzle analysis, counter-examples

angle relationships

angle relationships

- properties, proofs, parallel lines, triangles and other polygons, angle constructions

graphical analysis:

graphical analysis

- using technology only

- linear inequalitieslinear inequalities
- graphing of the solution region
- slope and intercepts
- intersection points of lines

- quadratic functionsquadratic functions
- characteristics of graphs, including end behaviour, maximum/minimum, vertex, symmetry, intercepts

- systems of equationssystems of equations
- including linear with linear, linear with quadratic, and quadratic with quadratic

- optimizationoptimization
- using feasible region to optimize objective function
- maximizing profit while minimizing cost
- maximizing area or volume while minimizing perimeter

applications of statistics

applications

- posing a question about an observed variation, collecting and interpreting data, and answering the question

statistics

- measures of central tendency, standard deviation, confidence intervals, z-scores, distributions

scale models

scale models

- enlargements and reductions of 2D shapes and 3D objects
- comparing the properties of similar objects (length, area, volume)
- square-cube law

financial literacy: compound interest, investments and loans

financial literacy

- compound interest
- introduction to investments/loans with regular payments using technology
- buy/lease

**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.