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-Understanding of numbers is "Unitary" (no grouping by ten)
-Grades k-2 no concept of 10 and more

-Understanding "ten and some more" results in a number from 11-20.

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Ability to discern between quantities as having more, less or same.

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Connecting quantity to abstract symbol.

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-Echo counting, without number sense (memorizing)
-Connecting number words to objects being counted/quantity

STAGES OF COUNTING:
1. Acoustic Counting - saying numbers but not understanding their connection to quantity
2. Asynchronous Counting - they understand numbers represent quantity without one-to-one correspondence
3. Synchronous Counting: counting quantity with one-to-one correspondence.
4. Resultative Counting: counting quantity synchronously with principles of stable order and cardinality.
5. Shortened Counting: understand the representations of number quantity, see it and count on from there. (Seeing a set of 3 dots, counting 3 and continuing to count additional single dots - 4,5,6, etc). This is a foundation for flexible counting skills

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The ability to apply a nuanced and integrated understanding of what numbers mean, how they relate to each other, and the patterns they generate.

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Understanding Quantity and Size of numbers

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-See the representational patterns of quantity (in dots or with manipulatives)
-Associate those patterns with the number symbol

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basic relationship of quantity when comparing items.

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Mapping Skills: Connect quantity with symbol

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-Understanding patterns created by numbers

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Understanding the quantity represented by numbers 1-10.

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-Understanding relational quantity of one or two more/less than a between numbers 1-10
-Later applied to two digit numbers (8 is 2 more than 6 = 18 is 2 more than 16)

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Understand quantitative relationships between a given number and 5 or 10

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-Understanding that a number can be made of different quantitative parts
-Understanding what those quantitative parts are (8 is made up of 2 and 6, 3 and 5, 4 and 4, 1 and 7)

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Understanding the quantities represented by numbers 10-20

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-Understanding "ten and some more" results in a number from 11-20.

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--Conservation of numbers: numbers stay the same regardless of how they are arranged
Principles:
--One to one correspondence:(counting an object in a group once - with one touch)
--Stable Order Principle:
"number words always progress in the same order" (Flores, 2018, p.30)
--Cardinality:
know that the last number said is the quantity of the set.
--Irrelevance Principle: understanding that you can count objects in any order (left - right, right to left - center to edge etc) and it does not change the final count.
--Abstraction Principle:
occurs when a person can apply all of the previous principles regardless of "what the collection of objects look like." (Flores, 2018, p.30)

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By Counts by Ones
*When presented with a large set of blocks and told to count*
1. Count set by ones
2. Count set with a mixture of groups and singles
3. Count set by groups of tens and ones
Viewing "ten" as ten ones grouped together.

Groupings with words
-Base Ten Language:
"4 tens and 2 ones is forty-two"
"3 groups of ten and 5 left over is thirty-five"

Groupings with Place Value
Seeing "10 ones" as a "set of 10" or "a 10" means
-Develop the idea of what "a ten" means using manipulatives

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-Relationship between a number added to itself and its sum.
-Using this understanding as a strategy for basic math facts

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-Relate numbers to measurements
-Understand what is meant by "about"
-Applying that understanding through measurements
-Collecting data and graphing data

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Ability to count forward sequentially and reverse that count sequentially

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Grouping by tens + connecting place value scheme, how numbers are written and spoken

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-Construct the idea of "a ten" and connect that with the model-

Models:

Proportional in size 1:10:100

"Put together and Take apart models"
-Interlocking Cubes, Counters and Cups, Sticks/Bundles

Pre-Grouped Models
-Base 10 blocks, Strips and squares, Ten-Frames

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Grouping tens to make hundreds

Develop understanding of "Hundreds" as:
-Singular units
-Groups of tens
-As a whole set

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-For addition:

Pre-reqs: counting, number sense, awareness of symbolic number representation

Join concrete objects, develop number sense

Introduce symbols later to learn "language of mathematics"

Understanding "properties" of addition

Use patterns to count (doubles, doubles plus one)

-For subtraction:

Pre-reqs: counting, number sense, awareness of symbolic number representation

Focus on conceptual development: separating/taking apart groups

Understanding it is the inverse of addition

Counting forward and backward

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Counting gradually through:
-Concrete manipulatives
-Semi-concrete/representational
-Counting up
-Count by grouping
-Counting abstractly - "Basic math fact"

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Practice:
-With base 10 mats
-Understand concepts before moving towards abstract procedural practice

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requires know-how of addition

Moving from counting strategies to reasoning strategies

Concepts of parts and whole/relationship between addition and subtraction

Reasoning Strategies:
Think addition - "What plus a equals b
Down under 10 - reverse of making 10, using 10 as an anchor

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Process:
-Concrete manipulatives
-Semi-concrete/representational
-Counting down
-Counting abstractly - "Basic math fact"

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-Subtraction by Counting Up
-Subtraction by Taking Away

-Move from open ended invented strategies to traditional algorithms

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Begin with Models - concrete representations, like base ten sheets or ten frames (especially for borrowing)

Trouble with subtracting zero - tackle during the model phase to teach borrowing

Writing the process - teach to write down all the steps in solving the problem

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-For Multiplication

Pre-reqs: Understanding of joining of equal groups/multiple addition

Understand that number symbols represent quantities of objects or groups of objects

Understand Place Value and base 10 system and understand the relationships between these numbers

-For Division

Pre-reqs: Understanding of subtraction

separating equal groups of objects

numbers make up equal groups

multiplication and division are inverse operations

knowledge of multiplication builds on skills in division

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Multiplication:

Memorizing facts: not very helpful - cognitive overload

Visual representation strategies: drawing out "objects" and grouping them

Multiple Addition

Breaking Numbers down/using distributive property

Division:

Visual representation: drawing objects and dividing them into equal groups

Repeated subtraction

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Reasoning Strategies for Multiplication and Division

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Reasoning Strategies for Multiplication and Division

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Foundational Facts: skip counting and doubles (0, 1, 2, 5, 10)

Teaching 9's: using 10 as an anchor (5 x 10 = 50, remove a set of 5 and you get 5 x 9 = 45)

Remaining facts and strategies:

Using Foundation facts as reference points to solve facts with nearby numbers

Add or Subtract a group - 6 x 4 can be solved by doing 5 x 4 and then adding another group of 4.

Doubling and Halving:

Different combinations of repeated addition using the doubling principle. Double and double again, double and add a set or half then double strategies

Break apart: decompose bigger number into smaller addends, multiply by other number and then add sums together.

Use these multiplication strategies to inform the inverse solution for related division problems

Division - Traditional Methods

Single number division - work on recognizing place value in context of division

Two digit division (out dated for mastery - since in context, calculators are available)

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-Fractions are a Part/Whole Value (Partitioning)

Fraction Notation and Vocabulary (numerator, denominator, fourth, halves, etc)

Concrete understanding through area, set, and length models

Partitioning with fractional models

Fraction size is relative - it's the relationship between a part of the whole

Fractions are *equal parts* of a whole

-Fractions as Numbers (still partitioning)

As number that is a part of a whole numbers

As numbers less than one

As numbers greater than one

-Fractions as Constructs

Part-Whole

Measurement

Division

Operator (fraction OF a whole number 4/5 of 20)

Ratio (probability)

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Iteration of Fractions

Counting fractions

Numerator counts, Denominator is what is being counted

Deepens concept of "fractions as numbers"

length models and iteration

Equivalent Fractions

Understand within context using models

Concept goal: A fixed quantity can have more than one name

Area, Length and Set models can be used

Comparing Fractions/Estimating with Fractions

Thinking about fraction size:
-Same size whole
-Same size numerator/different whole (same number of parts)
-More/less than half or one
-Closeness to half/one

Using equivalent fractions to compare

Using benchmark numbers to estimate (compare to 0, 1/2, 1)

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Addition and Subtraction with like Denominators and Multiplying by whole numbers

Using area, set, and length models

Using iteration models

Like denominators

Unlike denominators

Multiply by whole numbers

Addition/Subtraction with Fraction Fluency and deepen understanding of multiplication/introduction of division

Invented methods - developing algorithms

Multiply whole numbers by a fraction

Subdivision of parts

Understanding of Division with Fractions

Whole number divided by Whole number

Fraction Divided by Whole number

Whole numbers divided by Fractions

Fractions divided by fractions

Apply procedural skills to real life context