Big Ideas
Big Ideas
Proportional reasoning is used to make sense of multiplicative relationships.
- reasoning about comparisons of relative size or scale instead of numerical difference
- the multiplicative relationship between two numbers or measures is a relationship of scale rather than an additive difference (e.g., “12 is three times the size of 4” is a multiplicative relationship; “12 is 8 more than 4” is an additive relationship)
- Sample questions to support inquiry with students:
- What are the similarities and differences between strategies for solving proportional reasoning problems in different contexts?
- How does understanding the relationship between multiplication and division help when working with proportions?
- How are proportions used to describe changes in size?
3D objects can be examined mathematically by measuring directly and indirectly length, surface area, and volume.
- Sample questions to support inquiry with students:
- What measurement is the most important for examining 3D objects?
- Why is it important to understand the components of a formula?
Flexibility with number builds meaning, understanding, and confidence.
- Sample questions to support inquiry with students:
- How does using a measuring tool increase fluency and flexibility with decimals and fractions?
- How does solving puzzles and playing games help our understanding of number?
- Why are fractions important for imperial measurements?
- How does base 10 make the metric system easier to use?
- How is the order of operations connected to formula calculations?
- How do we determine which unit is the most appropriate to use?
- What level of estimation is considered reasonable when purchasing goods?
Representing and analyzing data allows us to notice and wonder about relationships.
- Sample questions to support inquiry with students:
- How do we choose the most appropriate graph to represent a set of data?
- How do graphs help summarize and analyze data?
- How can simulations help us make inferences?
- How can investigating trends help us make predictions?
- Why are graphs used to represent data?
- Why do we graph data?
Content
Learning Standards
Content
create, interpret, and critique graphs
- including a variety of formats, such as line, bar, and circle graphs, as well as histograms, pictographs, and infographics
primary trigonometric ratios
- single right-angle triangles; sine, cosine, and tangent
metric and imperial measurement and conversions
- with a focus on length as a means to increase computational fluency
- using tools and appropriate units to measure with accuracy
surface area and volume
- including prisms and cylinders, formula manipulation
- contextualized problems involving 3D shapes
central tendency
- analysis of measures and discussion of outliers
- calculation of mean, median, mode, and range
experimental probability
- simulations through playing and creating games and connecting to theoretical probability where possible
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 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 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 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
- 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
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 (www.fnesc.ca/wp/wp-content/uploads/2015/09/PUB-LFP-POSTER-Principles-of…) ( 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 (http://www.csus.edu/indiv/o/oreyd/ACP.htm_files/abishop.htm): counting, measuring, locating, designing, playing, explaining
- Aboriginal Education Resources (www.aboriginaleducation.ca)
- Teaching Mathematics in a First Nations Context, FNESC (http://www.fnesc.ca/resources/math-first-peoples/)