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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 x
^{2}+bx+c extend to ax^{2}+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 ax
^{2}+ bx + c - difference of squares of the form a
^{2}x^{2}- b^{2}y^{2} - may extend to a(f(x))
^{2}+ b(f(x)) +c, a^{2}(f(x))^{2}- b^{2}(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, x
^{2}- 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.