# data[].attributes.body
Furthering mathematical understanding by asking questions and using resources.

Examples:

Explore different methodologies

Research relevant ideas

https://nrich.maths.org/2293

# data[].attributes.attachment_link.readable_text
It's easy to work out the areas of squares drawn on a grid if they are oriented in the usual way:

Can you find a quick and easy method to work out the areas of tilted squares?

Here are some squares with a tilt of 1:

See the hint for suggested ways to calculate their areas.

Notice anything special about their areas?
Can you predict the areas of other squares with a tilt of 1?

What about squares with a tilt of 2? Or 3? Or 4? Or...?
Notice anything interesting?

Can you make any conjectures about the areas of tilted squares?
Can you prove your conjectures?

You might like to use the interactivity below to help you to draw tilted squares.

Full Screen tablet/mobile friendly version

# data[].attributes.attachment_link.keywords[]
md angles polygons and geom proof - stage 3

# data[].attributes.attachment_link.keywords[]
pm - posing questions. making conjectures

# data[].attributes.attachment_link.description
It's easy to work out the areas of squares drawn on a grid if they are oriented in the usual way: Can you find a quick and easy method to work out the areas of tilted squares? Here are some squares with a tilt of 1: See the hint for suggested ways to calculate their areas.

# data[].attributes.body
Helps organize, record, and communicate ideas

Models and interprets a variety of situations

Examples:

Use words, drawings, tables, charts, graphs, diagrams

Include multiple representations

# data[].attributes.body
Expand understanding to related concepts because math is a cumulative body of knowledge.

Make connections...

between facts

between symbols and procedures

to the real world

to old problems

# data[].attributes.body
Talking and writing about mathematics.

Examples:

Listen & discuss as a group

Use correct mathematical language

Explain ideas

# data[].attributes.body
The ability to make generalizations and provide explanations and justification for arguments.

Examples:

Use data to make a conjecture

Ask “why”

Recognize mistakes

# data[].attributes.body
Thinking logically and making decisions in a systematic way.

1. Understand the problem
2. Devise a plan
3. Carry out the plan
4. Review and extend

Examples:

Look for a pattern

Guess and check

Work backwards

Solve a simpler/similar problem

# data[].attributes.body
Each category can be broken down into strategies and skills.

1. Problem Solving
2. Reasoning and Proof
3. Communicating
4. Making Connections
5. Creating Representations
6. Investigating

# data[].attributes.body
A set of computational rules to produce a specified outcome.

Consist of skills

Memorizing an algorithm does not imply conceptual understanding

Example: PEMDAS (parentheses, exponent, multiplication, division, addition, subtraction)

# data[].attributes.body
Small operations/actions that are embedded in strategies.

Allow strategies to work when applied appropriately

Examples: being able to create a table of values, plot a graph, or use trial and error to solve a problem

# data[].attributes.body
A number of skills that learners use in a methodical and systematic way to support learning.

# data[].attributes.body
Broad techniques that support the understanding of concepts.

Composed of strategies, algorithms, and skills

Six categories

# data[].attributes.body
Vocabulary, definitions, mathematical formulae in symbol form.

Rote memorization

Does not imply conceptual understanding

# data[].attributes.body
"Broader" concepts: micro concepts.

# data[].attributes.body
Give supportive depth of understanding.

Examples: ratio, angles, similarity

# data[].attributes.body
Broken down macro concepts.

Examples: trigonometry, functions, differentiation

# data[].attributes.body
Give breadth of understanding and have greatest transferability across most disciplines.

Examples: geometry, algebra, calculus

# data[].attributes.body
Generalizations and principles are statements of conceptual understanding that allow students to make connections between two or more concepts

Principles are theorems

Theorems: statements that have been proven and connect explanations of conceptual understandings

# data[].attributes.body
Timeless

Universal

Transferable

Abstract to varying degrees

One or two words or a short phrase

Have different examples that share common attributes