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# Workplace Mathematics 11

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### Big Ideas

### Grandes idées

Proportional reasoning is used to make sense of multiplicative relationships.

Proportional reasoning

- reasoning about comparisons of relative size or scale instead of numerical difference

multiplicative

- 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:*- How are proportions used to describe changes in size?
- How are proportions used to solve problems in different contexts?
- How can proportions be used to represent and analyze rates of change?
- As the proportions of a shape change, what happens to the angles?

Mathematics informs financial decision making.

decision making

*Sample questions**to support inquiry with students:*- How do we make informed financial decisions?
- What factors should be considered when making a large purchase?
- What are the benefits of making responsible financial decisions?

3D objects are often represented and described in 2D space.

3D objects

*Sample questions**to support inquiry with students:*- Why is it important to represent 3D objects on a 2D plane?
- Where are representations of 3D objects used outside the classroom?
- Why is accuracy of measurement important when looking at scale diagrams?
- Can all 3D objects be described using 2D representations?
- What do we notice about angles in scale diagrams?

Flexibility with number builds meaning, understanding, and confidence.

understanding

*Sample questions to support inquiry with students:*- How does solving puzzles and playing games relate to mathematics?
- How does experiential learning facilitate deeper understanding?

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

notice and wonder

*Sample questions to support inquiry with students:*- How can statistical analysis help us make inferences about the future?
- How can a trend be determined from a set of given data?
- How can mathematics be used to influence our decisions around positive changes in society?

## 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., rate of change, trigonometry calculations)

reason

- inductive and deductive reasoning
- predictions, generalizations, conclusions drawn from experiences (e.g., with puzzles, games, coding)

technology

- graphing technology, dynamic geometry, calculators, virtual manipulatives, concept-based apps
- can be used for a wide variety of purposes, including:
- 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., trigonometric angle/side relations and rate of change calculations)

fluent, flexible, and strategic thinking

- includes:
- using known facts and benchmarks and partitioning (e.g., creating and interpreting 3D diagrams and making financial decisions based on evidence)
- 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

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, experimental, 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/)

**connections with mathematical concepts**### Content

*Students are expected to know the following:*financial literacy: personal investments, loans, and budgeting

financial literacy

- personal investments, loans (lease versus buy), credit cards, mortgages, graphical representations of financial growth
- to purchase, own, or lease and to operate and maintain a vehicle
- banking services
- other significant purchases

rate of change

rate of change

- slope of 3D objects, angle of elevation
- interest rates

how probability and statistics are used in different contexts

contexts

- exploring games of chance and insurance payout likelihood
- reading about and interpreting surveys and information in the media to make informed decisions
- understanding statistical vocabulary

interpreting graphs in society

interpreting graphs

- investigating graphs in the media (e.g., news articles, blogs, social media, websites, advertisements)
- how data and media influence social justice issues and personal decisions

3D objects: angles, views, and scale diagrams

3D objects

- creating and interpreting exploded diagrams and perspective diagrams
- drawing and constructing 3D objects

**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.