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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.