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Pre-calculus 11
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Big Ideas
Grandes idées
Algebra allows us to generalize relationships through abstract thinking.
generalize
- Sample questions to support inquiry with students:
- After solving a problem, can we extend it? Can we generalize it?
- How can we take a contextualized problem and turn it into a mathematical problem that can be solved?
- How do we tell if a mathematical solution is reasonable?
- Where can errors occur when solving a contextualized problem?
- What are the similarities and differences between quadratic functions and linear functions? How are they connected?
- What do we notice about the rate of change in a quadratic function?
- How do the strategies for solving linear equations extend to solving quadratic, radical, or rational equations?
- What is the connection between domain and extraneous roots?
The meanings of, and connections between, operations extend to powers, radicals, and polynomials.
connections
- Sample questions to support inquiry with students:
- How are the different operations (+, -, x, ÷, exponents, roots) connected?
- What are the similarities and differences between multiplication of numbers, powers, radicals, polynomials, and rational expressions?
- How can we verify that we have factored a trinomial correctly?
- How can visualization support algebraic thinking?
- How can patterns in numbers lead to algebraic generalizations?
- When would we choose to represent a number with a radical rather than a rational exponent?
- How do strategies for factoring x2+bx+c extend to ax2 +bx + c, a≠1
- How do operations on rational numbers extend to operations with rational expressions?
Quadratic relationships are prevalent in the world around us.
relationships
- Sample questions to support inquiry with students:
- What are some examples of quadratic relationships in the world around us, and what are the similarities and differences between these?
- Why are quadratic relationships so prevalent in the world around us?
- How does the predictable pattern of linear functions extend to quadratic functions?
- Why is the shape of a quadratic function called a parabola?
- How can we decide which form of a quadratic function to use for a given problem?
- What effect does each term of a quadratic function have on its graph?
Trigonometry involves using proportional reasoning to solve indirect measurement problems.
proportional reasoning
- comparisons of relative size or scale instead of numerical difference
indirect measurement
- using measurable values to calculate immeasurable values (e.g., calculating the width of a river using the distance between two points on one shore and an angle to a point on the other shore)
- Sample questions to support inquiry with students:
- How is the cosine law related to the Pythagorean theorem?
- How can we use right triangles to find a rule for solving non-right triangles?
- How do we decide when to use the sine law or cosine law?
- What would it mean for an angle to have a negative measure? Identify a context for making sense of a negative angle.
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., trinomial factoring, roots of quadratic equations)
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., the zeros of a graphed polynomial function)
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 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 effective 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, with other areas, and with 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:
real number system
real number
- classification
powers with rational exponents
powers
- positive and negative rational exponents
- exponent laws
- evaluation using order of operations
- numerical and variable bases
radical operations and equations
radical
- simplifying radicals
- ordering a set of irrational numbers
- performing operations with radicals
- solving simple (one radical only) equations algebraically and graphically
- identifying domain restrictions and extraneous roots of radical equations
polynomial factoring
factoring
- greatest common factor of a polynomial
- trinomials of the form ax2 + bx + c
- difference of squares of the form a2x2 - b2y2
- may extend to a(f(x))2 + b(f(x)) +c, a2(f(x))2 - b2(f(x))2
rational expressions and equations
rational
- simplifying and applying operations to rational expressions
- identifying non-permissible values
- solving equations and identifying any extraneous roots
quadratic functions and equations
quadratic
- identifying characteristics of graphs (including domain and range, intercepts, vertex, symmetry), multiple forms, function notation, extrema
- exploring transformations
- solving equations (e.g., factoring, quadratic formula, completing the square, graphing, square root method)
- connecting equation-solving strategies
- connecting equations with functions
- solving problems in context
linear and quadratic inequalities
inequalities
- single variable (e.g., 3x - 7 ≤ -4, x2 - 5x + 6 > 0)
- domain and range restrictions from problems in situational contexts
- sign analysis: identifying intervals where a function is positive, negative, or zero
- symbolic notation for inequality statements, including interval notation
trigonometry: non-right triangles and angles in standard position
trigonometry
- use of sine and cosine laws to solve non-right triangles, including ambiguous cases
- contextual and non-contextual problems
- angles in standard position:
- degrees
- special angles, as connected with the 30-60-90 and 45-45-90 triangles
- unit circle
- reference and coterminal angles
- terminal arm
- trigonometric ratios
- simple trigonometric equations
financial literacy: compound interest, investments, 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.