Math Learning Sequence by Gabrielle Wirth

Math Learning Sequence by Gabrielle Wirth https://padlet.com/gswirth/z6fgmi9qgwgm Concept Map by Gabrielle Wirth en-us 2020-01-29 12:43:53 UTC 2020-04-09 14:03:09 UTC hello@padlet.com Number Sense gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437501144 The ability to apply a nuanced and integrated understanding of what numbers mean, how they relate to each other, and the patterns they generate. ]]> 2020-01-29 12:48:45 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437501144 Code/Symbolic Representation gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437684295 Mapping Skills: Connect quantity with symbol

]]> 2020-01-29 17:00:49 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437684295 Non-Symbolic Representation gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437684908 Understanding Quantity and Size of numbers]]> 2020-01-29 17:01:35 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437684908 Counting and Quantity Knowledge gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437686550 2020-01-29 17:03:30 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437686550 See Relationships between numbers gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437687040 -Understanding patterns created by numbers]]> 2020-01-29 17:04:06 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437687040 Numbers 1-10 gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437688055 Understanding the quantity represented by numbers 1-10.

]]> 2020-01-29 17:05:20 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437688055 More, Less, Same gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437689874 basic relationship of quantity when comparing items.]]> 2020-01-29 17:07:16 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437689874 Recognize Patterns in Number Sets gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437693343 -See the representational patterns of quantity (in dots or with manipulatives)
-Associate those patterns with the number symbol ]]> 2020-01-29 17:12:00 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437693343 One or Two More/Less gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437699347 -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) ]]> 2020-01-29 17:19:52 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437699347 Anchoring Numbers to 5 and 10 gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437703099 Understand quantitative relationships between a given number and 5 or 10 ]]> 2020-01-29 17:23:58 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437703099 Part-Part-Whole Relationship gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437705055 -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) ]]> 2020-01-29 17:26:27 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437705055 Numbers 10-20 gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437707491 Understanding the quantities represented by numbers 10-20]]> 2020-01-29 17:29:41 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437707491 Pre-Place Value gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437712742 -Understanding "ten and some more" results in a number from 11-20. ]]> 2020-01-29 17:36:37 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437712742 Extending One or Two More/Less gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437714080 2020-01-29 17:38:23 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437714080 Doubles and Near Doubles gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437716827 -Relationship between a number added to itself and its sum.
-Using this understanding as a strategy for basic math facts]]> 2020-01-29 17:41:41 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437716827 Estimation and Measurement gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437721136 -Relate numbers to measurements
-Understand what is meant by "about"
-Applying that understanding through measurements
-Collecting data and graphing data]]> 2020-01-29 17:47:11 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/437721136 Counting Back and Counting On gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/438119272 Ability to count forward sequentially and reverse that count sequentially ]]> 2020-01-30 13:22:48 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/438119272 Numerical Recognition gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/438120000 Connecting quantity to abstract symbol.]]> 2020-01-30 13:24:09 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/438120000 Counting Principles gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/438120098 --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)

]]> 2020-01-30 13:24:21 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/438120098 More, Less, Same gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/438120577 Ability to discern between quantities as having more, less or same. ]]> 2020-01-30 13:25:04 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/438120577 Beginning Counting gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/442219326 -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

]]> 2020-02-07 15:55:26 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/442219326 Pre-Place Value gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449441536 -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. ]]> 2020-02-24 13:44:36 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449441536 Basic Base Ten gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449444630 Grouping by tens + connecting place value scheme, how numbers are written and spoken

]]> 2020-02-24 13:49:18 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449444630 Integration of Base Ten Grouping gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449445461

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

]]> 2020-02-24 13:50:28 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449445461 Using Base 10 Models gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449452599 -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

]]> 2020-02-24 14:01:17 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449452599 Tens - Hundreds gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449455627 Grouping tens to make hundreds

Develop understanding of "Hundreds" as:
-Singular units
-Groups of tens
-As a whole set ]]> 2020-02-24 14:05:58 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449455627 Single Digit Addition gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449457196 Counting gradually through:
-Concrete manipulatives
-Semi-concrete/representational
-Counting up
-Count by grouping
-Counting abstractly - "Basic math fact"]]> 2020-02-24 14:08:33 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449457196 Conceptual understanding gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449458888 -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

]]> 2020-02-24 14:11:15 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449458888 Addition with Re-grouping gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449461227 Practice:
-With base 10 mats
-Understand concepts before moving towards abstract procedural practice

]]> 2020-02-24 14:15:11 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449461227 Subtraction - Understanding/Reasoning gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449465422 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

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

]]> 2020-02-24 14:21:48 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449465422 Basic Subtraction gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449472309 -Subtraction by Counting Up
-Subtraction by Taking Away

-Move from open ended invented strategies to traditional algorithms

]]> 2020-02-24 14:31:54 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449472309 The Subtraction Algorithm gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449472458 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 ]]> 2020-02-24 14:32:08 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/449472458 Conceptual Understanding gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/473601691 -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

]]> 2020-03-24 21:29:43 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/473601691 Reasoning Strategies for Multiplication and Division gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/473610385 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)

]]> 2020-03-24 21:38:02 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/473610385 Invented Strategies gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/473644372 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

]]> 2020-03-24 22:11:14 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/473644372 Conceptual Understanding gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/500486298 -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)

]]> 2020-04-09 13:13:01 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/500486298 Number Sense with Fractions gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/500553754 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)

]]> 2020-04-09 13:46:44 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/500553754 Procedural Understanding/Computing with Fractions gswirth https://padlet.com/gswirth/z6fgmi9qgwgm/wish/500563335 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

]]> 2020-04-09 13:51:23 UTC https://padlet.com/gswirth/z6fgmi9qgwgm/wish/500563335