Big Ideas
Big Ideas
Using inverses is the foundation of solving equations and can be extended to relationships between functions.
- undo the operations within an expression or function to reduce the expression to an identity (e.g., x = )
- Sample questions to support inquiry with students:
- How can the inverse help to solve an equation?
- How is solving an equation related to identifying the specific input for a function, given a specific output?
- How are exponential and logarithmic functions related?
- How are the laws of exponents connected to the laws of logarithms?
- What are some other examples of inversely related functions?
- How are inverses related graphically, and why?
- How is solving an exponential equation similar to solving a trigonometric equation?
- How are inverse operations related to solving a polynomial equation by factoring?
- What is the value of using trigonometric identities to find equivalent expressions?
- Why do some equations have extraneous roots and other equations do not?
Understanding the characteristics of families of functions allows us to model and understand relationships and to build connections between classes of functions.
- Sample questions to support inquiry with students:
- How do we decide which kind of function to use to model a given problem?
- What do functions and relations look like beyond the visible axes?
- A set of data looks like a parabola, but it is not. What function could be used to model this data?
- What does the number of zeros tell us about a function?
- What connections do we see within the characteristics of a particular class of function?
Transformations of shapes extend to functions and relations in all of their representations.
- Sample questions to support inquiry with students:
- How can we tell whether a transformation will have invariant points?
- Under what circumstances will different transformations produce the same result?
- How do graphical transformations affect the tables of values?
- How does a transformation affect a point found at the origin as compared to a point on an axis or a point in one of the four quadrants?
- How can a rational function of the form y = (ax+b) / (cx+d) be considered as a transformation of the reciprocal function y = 1/x ?
Content
Learning Standards
Content
transformations of functions and relations
- of graphs and equations of parent functions and relations (e.g., absolute value, radical, reciprocal, conics, exponential, logarithmic, trigonometric)
- vertical and horizontal translations, stretches, and reflections
- inverses: graphs and equations
- extension:
- recognizing composed functions (e.g., y =)
- operations on functions
exponential functions and equations
- graphing, including transformations
- solving equations with same base and with different bases, including base e
- solving problems in situational contexts
geometric sequences and series
- common ratio, first term, general term
- geometric sequences connecting to exponential functions
- infinite geometric series
- sigma notation
logarithms: operations, functions, and equations
- applying laws of logarithms
- evaluating with different bases
- using common and natural logarithms
- exploring inverse of exponential
- graphing, including transformations
- solving equations with same base and with different bases
- solving problems in situational contexts
polynomial functions and equations
- factoring, including the factor theorem and the remainder theorem
- graphing and the characteristics of a graph (e.g., degree, extrema, zeros, end-behaviour)
- solving equations algebraically and graphically
rational functions
- characteristics of graphs, including asymptotes, intercepts, point discontinuities, domain, end-behaviour
trigonometry : functions, equations, and identities
- examining angles in standard position in both radians and degrees
- exploring unit circle, reference and coterminal angles, special angles
- graphing primary trigonometric functions, including transformations and characteristics
- solving first- and second-degree equations (over restricted domains and all real numbers)
- solving problems in situational contexts
- using identities to reduce complexity in expressions and solve equations (e.g., Pythagorean, quotient, double angle, reciprocal, sum and difference)
Curricular Competency
Learning Standards
Curricular Competency
Reasoning and modelling
Develop thinking strategies to solve puzzles and play games
- using reason to determine winning strategies
- generalizing and extending
Explore, analyze, and apply mathematical ideas using reason, technology, and other tools
- examine the structure of and connections between mathematical ideas (e.g., exponential functions to geometric sequences)
- inductive and deductive reasoning
- predictions, generalizations, conclusions drawn from experiences (e.g., with puzzles, games, and coding)
- graphing technology, dynamic geometry, calculators, virtual manipulatives, concept-based apps
- can be used to for a wide variety of purposes, including:
- exploring and demonstrating mathematical relationships
- organizing and displaying data
- generating and testing inductive conjectures
- mathematical modelling
- manipulatives such as algebra tiles and other concrete materials
Estimate reasonably and demonstrate fluent, flexible, and strategic thinking about number
- 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)
- 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
- 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
- including real-life scenarios and open-ended challenges that connect mathematics with everyday life
Think creatively and with curiosity and wonder when exploring problems
- 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
- 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
- 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
- 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
- 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)
- 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
- 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
- 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
- use mathematical arguments to convince
- includes anticipating consequences
- Have students explore which of two scenarios they would choose and then defend their choice.
- including oral, written, visual, use of technology
Represent mathematical ideas in concrete, pictorial, and symbolic forms
- using models, tables, graphs, words, numbers, symbols
- connecting meanings among various representations
Use mathematical vocabulary and language to contribute to discussions in the classroom
- partner talks, small-group discussions, teacher-student conferences
Take risks when offering ideas in classroom 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
- 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
- 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
- range from calculation errors to misconceptions
- 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
- 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-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 mathematics
- exploring cultural practices and knowledge of local First Peoples and identifying mathematical connections
- local knowledge and cultural practices that are appropriate to share and that are non-appropriated
- 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/)