# data[].attributes.body
1. Positional notation
2. Piles and Holes

# data[].attributes.body
1. Tally
2. Symbols
OTHER: Some first figurate numbers
How we count: Feynman reading story
James' Horse Counting?

# data[].attributes.body
* Develop topic strands
* Not every strand need begin at grade 3. Sort out progressions here.

# data[].attributes.body
Place Value (PNS)
Multiplication (Aaron and Loretta)
Fractions (Hannah and Joan)
Subtraction

# data[].attributes.body
Specify what we are presuming as pre-existing understanding and skill sets.

# data[].attributes.body
* Need to mull on this.
Even though many curricula do this early on, I am not sure what problem the notion of a function is solving for the student right now. (I fear this is another "Do this now because you will need to know it later on" thing.)

# data[].attributes.body
* Central tendency
* variation
* distributions
* sampling
* inference
* linear regression (Connect with equations of lines)

# data[].attributes.body
* Review of garden paths
* Believing certain scenarios come with inherent probability values. (Coin: p = 1/2. Die: p = 1/6, etc.)
* Using data to estimate p

# data[].attributes.body
* Square roots
* Pythagorean Theorem
* More on the coordinate plane.
* Distance in 2 dimensions. (Connect to distance in 1 dimension)
* Circles and pi
* Parallelism
* Rigid motions
* Dilations
* similarity and congruence
* What we choose to believe about area and volume

# data[].attributes.body
* Positive integer exponents
* What are we choosing to believe?
* zeroth powers, negative powers
* square roots as powers
* Scientific Notation

# data[].attributes.body
* Radically decluttered version of all this!

# data[].attributes.body
* All the usual
* Existence of irrational numbers

# data[].attributes.body
* Usual horrors of understanding, graphing, solving
* Pairs of linear equations
* Connect with linear regression when time comes

# data[].attributes.body
* Deep understanding
* Simultaneous equations solved graphically

# data[].attributes.headline
From 1 dimension to 2: Coordinate Systems

# data[].attributes.subject
From 1 dimension to 2: Coordinate Systems

# data[].attributes.body
* The power of two number lines
* Scatter plots for display of data

# data[].attributes.body
* Deep understanding of this
* The art of solving equations

# data[].attributes.body
* For us, that language is English!
* Writing mathematical sentences (with punctuation!)
* =, > , < , and so on.
* Meaning of these in terms of balance scales:
- intuition for basic properties

# data[].attributes.body
* Focus on unit fractions 1/2, 1/3, /14, ... and their key property: n x 1/n = 1
* Define a/b to be a copies of 1/b:   a/b = a * 1/b.
* a/b is the answer to the division problem a ÷b. so "deserve" number status.
* Defining ÷, x, +, - for fractions
* All the usual fraction content to be covered
* Recognizing (-a)/b and a/(-b) and - a/b as equivalent.
* Placing fractions on the number line.
* Distance on the number line (aka absolute value)

# data[].attributes.body
* Practice writing, creating, evaluating mathematical expressions
* Identifying equivalent expressions

# data[].attributes.body
* Review what we "know" about the system 0, 1, 2, 3, 4, ....
* Identify basic properties of numbers that feel fundamental and "right."
* Do we want the opposite numbers to follow the same rules?
Logical consequences. (e.g. negve x negve = posve)
* Area model extended

# data[].attributes.body
* Fun with fractional thinking
* Seeds of probability theory

# data[].attributes.body
* A focus on troublesome zero
* Opposite numbers (negative numbers)
* Might fractions actually be numbers? (Review of the models and what they hint at)
* The number line

# data[].attributes.body
* units, metric system
* basic unit conversion
* time
* Data collection and display

# data[].attributes.body
* Shapes and their characteristics
* Area. Perimeter
* Volume
* Angles
* Symmetry

# data[].attributes.body
* Introduction
* Rounding

# data[].attributes.body
* First models
* addition, subtraction, "of", equivalence, comparison
* mixed numbers
* Basic division: 1/2 ÷ 3
* Hints of the number line

# data[].attributes.body
* Multiplication (and division?)
* Chopping up rectangles (distributive rule)

# data[].attributes.body
* Square and Triangular Numbers
* Mathematical Expressions - seeds for algebraic expressions
* Prime and Composite numbers --> clever multiplying
* Even/Odd numbers: Parity

# data[].attributes.body
* Exploding Dots (actually ... for stage 2)
* Rounding
* Estimation

# data[].attributes.body
* The decimal system
* Multiplication and Division
* Subtraction
* Order of operations  (vinculum/parentheses)

# data[].attributes.body
* First evidence of counting
* The counting numbers and zero
* Addition and "take away" -- mentions
* Mention even and odd numbers.
* (* Jo Boaler's number play can be good meta-messaging here)