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# Foundations of Mathematics and Pre-calculus 10

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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 can we tell if a mathematical solution is reasonable?
- Where can errors occur when solving a contextualized problem?
- What do we notice when we square binomials?
- How do we decide on a strategy for solving a system of equations?

The meanings of, and connections between, each operation extend to powers and polynomials.

connections

*Sample**questions to support inquiry with students:*- How are the different operations (+, -, x, ÷, exponents) connected?
- What are the similarities and differences between multiplication of numbers, powers, and polynomials?
- How is prime factorization helpful?
- How does prime factorization of numbers extend to algebraic terms?
- 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?

Constant rate of change is an essential attribute of linear relations and has meaning in different representations and contexts.

relations

*Sample**questions to support inquiry with students:*- How can we tell if a relation is linear?
- How can we use rate of change to make predictions?
- What connections can we make between arithmetic sequences and linear functions?
- How do we decide which form of linear equation to use?

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 height of a tree using distance from the tree and the angle to the top of the tree)
*Sample questions to support inquiry with students:*- When might we need to measure a length or angle indirectly?
- Why is trigonometry defined in reference to right triangles rather than other types of triangles?
- How can rate of change be connected to trigonometry?
- What is the origin of the names for the trigonometric ratios?

Representing and analyzing situations allows us to notice and wonder about relationships.

situations

- situational contexts (e.g., relating volume to height when filling containers of different shapes, relating distance to time for a bike ride)
- non-situational contexts (e.g., the graph of a piecewise function)
*Sample questions to support inquiry with students:*- How does the representation of a relation support a strategy when solving a problem?
- Do all data have trends and relationships?
- Why are trends important?

## Learning Standards

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### Curricular Competencies

### 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., using an area model to factor a trinomial)

reason

- inductive and deductivereasoning
- 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 to 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., estimating the solution for a system of equations from a graph)

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
- using concrete materials and dynamic interactive technology

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

operations on powers with integral exponents

powers

- positive and negative exponents
- exponent laws
- evaluation using order of operations
- numerical and variable bases

prime factorization

prime factorization

- expressing prime factorization of a number using powers
- identifying the factors of a number

- includes greatest common factor (GCF) and least common multiple (LCM)
- strategies include using factor trees and factor pairs

functions and relations: connecting data, graphs, and situations

functions and relations

- communicating domain and range in both situational and non-situational contexts
- connecting graphs and context
- understanding the meaning of a function
- identifying whether a relation is a function
- using function notation

linear functions: slope and equations of lines

linear functions

- slope: positive, negative, zero, and undefined
- types of equations of lines (point-slope, slope intercept, and general)
- equations of parallel and perpendicular lines
- equations of horizontal and vertical lines
- connections between representations: graphs, tables, equations

arithmetic sequences

arithmetic sequences

- applying formal language (common difference, first term, general term) to increasing and decreasing linear patterns
- connecting to linear relations
- extension: exploring arithmetic series

systems of linear equations

systems

- solving graphically
- solving algebraically by inspection, substitution, elimination
- connecting ordered pair with meaning of an algebraic solution
- solving problems in situational contexts

multiplication of polynomial expressions

multiplication

- applying the distributive property between two polynomials, including trinomials
- connecting the product of binomials with an area model

polynomial factoring

factoring

- greatest common factor of a polynomial
- simpler cases involving trinomials y = x
^{2}+ bx + c and difference of squares

primary trigonometric ratios

trigonometric

- sine, cosine, and tangent ratios
- right-triangle problems: determining missing sides and/or angles using trigonometric ratios and the Pythagorean theorem
- contexts involving direct and indirect measurement

financial literacy: gross and net pay

financial literacy

- types of income
- income tax and other deductions

**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.