Big Ideas
Big Ideas
Algebra allows us to generalize relationships through abstract thinking.
- 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.
- 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.
- 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.
- comparisons of relative size or scale instead of numerical difference
- 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.
- 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?
Content
Learning Standards
Content
operations on powers with integral exponents
- positive and negative exponents
- exponent laws
- evaluation using order of operations
- numerical and variable bases
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
- 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
- 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
- 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
- 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
- applying the distributive property between two polynomials, including trinomials
- connecting the product of binomials with an area model
polynomial factoring
- greatest common factor of a polynomial
- simpler cases involving trinomials y = x2 + bx + c and difference of squares
primary trigonometric ratios
- 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
- types of income
- income tax and other deductions
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., using an area model to factor a trinomial)
- inductive and deductivereasoning
- 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., estimating the solution for a system of equations from a graph)
- 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 mathematical understanding 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 appropriate 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
- communicating effectively according to what is being communicated and to whom
Represent mathematical ideas in concrete, pictorial, and symbolic forms
- 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
- 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/)