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# History of Mathematics 11

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### Big Ideas

### Grandes idées

Mathematics has developed over many centuries and continues to evolve.

developed

*Sample questions to support inquiry with students:*- What is the connection between the development of mathematics and the history of humanity?
- How have mathematicians overcome discrimination in order to advance the development of mathematics?
- Where have similar mathematical developments occurred independently because of geographical separation?

Mathematics is a global language used to understand the world.

language

*Sample questions to support inquiry with students:*- How universal is the language of mathematics?
- How is learning a language similar to learning mathematics?
- How does oral language influence our conceptual understanding of mathematics?

Societal needs across cultures have influenced the development of mathematics.

Societal needs

*Sample questions to support inquiry with students:*- Have societal needs always had a positive impact on mathematics?
- How have politics influenced the development of mathematics?
- How might mathematics influence decisions regarding social justice issues?

Tools and technology are catalysts for mathematical development.

Tools and technology

*Sample questions to support inquiry with students:*- Did tools and technology affect mathematical development or did mathematics affect the development of tools and technology?
- What does technology enable us to do and how does this lead to deeper mathematical understanding?

Notable mathematicians in history nurtured a sense of play and curiosity that led to the development of many areas in mathematics.

mathematicians

*Sample questions to support inquiry with students:*- What drives a mathematician to solve the seemingly unsolvable?
- What do you wonder about in the mathematical world?
- What are some examples of mathematical play that led to practical applications?

## Learning Standards

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### Curricular Competencies

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*Students are expected to be able to do the following:*### Reasoning and modelling

Develop thinking strategies to solve historical puzzles and play games

thinking strategies

- using reason to determine winning strategies
- generalizing and extending

Explore, analyze, and apply historical mathematical ideas using reason, technology,

analyze

- examine the structure of and connections between mathematical ideas from historical contexts

reason

- inductive and deductive reasoning
- predictions, generalizations, conclusions drawn from experiences

technology

- historically appropriate tools
- 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
- presenting historical solutions or mathematical ideas from a current perspective

**and other tools**other tools

- manipulatives such as rulers, compass, abacus, and other historically appropriate tools

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

Critique multiple strategies used to solve mathematical problems throughout history

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 problems (e.g., guess and check, model, solve a simpler problem, use a chart, use diagrams, role-play, historical representations)

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 and persevering through struggles (e.g., struggles of mathematicians and how their persistence led to mathematical discoveries)
- problem solving with vigour and determination

Engage in problem-solving experiences connected with place, story and cultural practices, including local First Peoples

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 argument 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

Use historical symbolic representations to explore mathematics

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, with other areas, and with 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)

Reflect on the consequences of mathematics culturally, socially, and politically

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:*number and number systems:

number and number systems

- Egyptian, Babylonian, Roman, Greek, Arabic, Mayan, Indian, Chinese, First Peoples
- exploring the idea of different bases, different forms of arithmetic
- infinity
- problems from the Rhind Mathematical Papyrus
- Eratosthenes

- written and oral numbers
- zero
- rational and irrational numbers
- pi
- prime numbers

patterns and algebra:

patterns and algebra

- Al-Khwarizmi’s
*Algebra* - Indian mathematics
- Islamic mathematics
- Descartes
- the golden ratio
- patterns in art

- early algebraic thinking
- variables
- early uses of algebra
- Cartesian plane
- notation
- Fibonacci sequence

geometry:

geometry

- problems from the Rhind Mathematical Papyrus, Moscow Mathematical Papyrus
- Pythagoras
- Hippocrates and construction problems of antiquity
- geometry in Euclid’s
*Elements*, Archimedes, Apollonius, Pappus’s*Book III* - Indian and Arabic contributions
- Descartes and Fermat

- of lines, angles, triangles
- Euclid’s five postulates
- geometric constructions
- developments through time

probability and statistics:

probability and statistics

- Pascal, Cardano, Fermat, Bernoulli, Laplace
- ancient games such as dice and the Egyptian game Hounds and Jackals
- Egyptian record keeping
- Graunt and the development of statistics through the need for merchant insurance policies

- Pascal’s triangle
- games involving probability
- early beginningsof statistics and probabilityearly beginnings
- forms of tabulating information, leading to the beginnings of probability and statistics

tools and technology: development over time, from clay tablets to modern-day calculators and computers

tools and technology

- papyrus, stone tablet, bone, compass and straightedge, abacus, scales, slide rule, ruler, protractor, calculator, computer

cryptography:

cryptography

- cuneiform
- Spartan military use of ciphers
- first documentation of ciphers in the Arab world
- John Wallis
- World War II and the Enigma machine
- barcodes
- modular arithmetic
- RSA coding
- current coding techniques and security in digital password encryption

- use of ciphers, encryption, and decryption throughout history
- modern uses of cryptography in war and digital applications

**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.