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# Apprenticeship Mathematics 12

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### Big Ideas

### Grandes idées

Design involves investigating, planning, creating, and evaluating.

Design

*Sample questions to support inquiry with students:*- How is a product designed?
- How can the design process be applied to meet a need or solve a problem?

Constructing 3D objects often requires a 2D plan.

3D objects

*Sample questions**to support inquiry with students:*- What are some limitations that result when 3D objects are represented in 2D?
- Which type of 2D representation would be the most appropriate for a 3D object?
- How does visualization help when solving problems?
- How does visualization help break down a larger problem?

Transferring mathematical skills between problems requires conceptual understanding and flexible thinking.

Transferring mathematical skills

*Sample questions**to support inquiry with students:*- How does awareness and knowledge of mathematics in the workplace make learning more meaningful?
- What is the mathematics required for a particular trade of interest?

Proportional reasoning is used to make sense of multiplicative relationships.

Proportional reasoning

- reasoning about comparisons of relative size or scale instead of numerical difference
- ways of showing proportional comparison when analyzing problems in situational contexts
- scale diagrams
- rates of change

*Sample**questions to support inquiry with students:*- How are proportions used to solve problems?
- What is the importance of proportional reasoning when making sense of the relationship between two things?

Choosing a tool based on required precision and accuracy is important when measuring.

measuring

*Sample questions to support inquiry with students:*- What skills are required for measuring with accuracy?
- What is the importance of choosing appropriate tools and units when measuring?
- What are the implications of inaccurate measurements?

## Learning Standards

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### Curricular Competencies

*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., proportional reasoning, metric/imperial conversions)

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 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 rulers and other measuring tools

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., reasonableness of measurements)

fluent, flexible, and strategic thinking

- including:
- using known facts and benchmarks, partitioning, applying whole number strategies to expressions involving proportional reasoning, financial analysis, and logic
- 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, 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

*Students are expected to know the following:*measuring: using tools with graduated scales; conversions using metric and imperial

measuring

- unit analysis
- precision and accuracy
- breaking of units into smaller divisions to get more precise measurements
- extension: project or presentation to share measurement concepts and skills used in a field/career of interest

similar triangles: including right-angle trigonometry

triangles

- situational examples such as stairs and roofs
- application of Pythagorean theorem
- situations involving multiple right-angle triangles

2D and 3D shapes: including area, surface area, volume, and nets

3D objects and their views (isometric drawing, orthographic projection)

3D objects

- creating and reading various types of technical drawings
- extension: project or presentation to share geometry concepts and skills used in a field/career of interest

mathematics in the workplace

mathematics in the workplace

- compare and contrast mathematics used in different workplace contexts
- interview someone working in a field of interest
- extension: project that includes an element of design and mathematical thinking

financial literacy: business investments and loans

financial literacy

- business investments, loans (lease versus buy), graphical representations of financial growth, projections, expenses
- extension: project or presentation to share mathematical concepts and skills used in a field/career of interest

**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.