Counting on Rules of Arithmetic CORA

Counting on Rules of Arithmetic.

We count on from a starting number like 3 by a second number, the count on number or shift number like 2.

3+2 means start at 3 and count on by 2.

We imagine we are counting on along a number line.  So first we review some number line rules.

  1. Zero is a unique tick on the number line.
  2. For each tick on the number line, there exists a unique tick immediately to the right of it.
  3. Zero is not a tick to the right of another tick.
  4. If the ticks to the right of two ticks are equal, then said two ticks are equal.
  5. If a set contains zero and each tick to the right of a tick, then it contains all the ticks on the number line.

Counting on Rules of Arithmetic Basic  CORAB

  1. Zero is a unique starting from number on the number line.
  2. For each tick on the number line, there exists a unique tick count on by one to the right of it.
  3. You can never count on by one and get to zero.
  4. If you count on by one from two ticks and get to the same tick, the two starting ticks were the same.
  5. If a set contains zero and you count on by one without ever stopping, you reach every natural number as you count on.

We introduce a special symbol for counting on by one.

1=0′

This is read count on from 0 by 1 gets you to 1 or is one.

2=1′.

Count on by one from 1 gets you to 2.

CORA definition of addition.

We now define addition of whole numbers in terms of counting on.  For us whole numbers include 0.

  1. If you count on zero from a starting number, you stay at the counting number.  In this case, we say the sum of the starting number and zero is the starting number.
  2. Suppose you start from an initial number, x, and count on by y.  Now start from the same initial number x and count on by y’.  Then you get to the same place as if you start at x+y and count on by one from there.

Examples of rule 1.

Start at 2 and count on by 0.  The result is 2.  You are at 2.  The sum of 2 and 0 is 2.  2+0 = 2.

Examples of Rule 2

Suppose that 2+3 = 5.

Now we want to count on from 2 by 4=3′.

2+3′ = 5′ = 6

So when we count on from 2 by 4 we get to the same starting at 5 and counting on by one.

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Now let’s hear you math ed panjandrums and nay-sayers oppose counting on.

Ha Ha Ha.

“counting on” addition kindergarten

http://kindergartencrayons.blogspot.com/2012/01/counting-on-some-are-catching-on.html

http://kdoublestuffed.blogspot.com/2011/03/counting-on-addition-dice.html

http://education.illinois.edu/smallurban/chancellorsacademy/documents/strategiesbygrade.pdf

==

Kindergarten
1st Grade
2nd Grade
Addition Counting on Combinations of 5 and 10 1 and 2 more 5 and 10 as an anchor Counting on Apply Properties of Operation (Commutative, Zero rule…) 1 and 2 more Doubles Combinations of 10 Decomposition/composing numbers to
1. Make a 10
2. Work with near doubles
3. Build off a known fact. Apply Properties of Operation (Commutative, Zero rule…) 1 and 2 more Doubles Combinations of 10 Decomposition/composing numbers to
1. Make a 10
2. Work with near doubles
3. Build off a known fact.
Subtraction Undoing Addition 1 and 2 less Counting up and down Think addition.
1. Counting-up
2. Reverse doubles.
3. Compliments of 10
4. Any subtraction fact Apply properties of operation 1 and 2 less Decompose/Recompose
1. Building up through 10
2. Building down through 10 Think addition.
1. Reverse doubles.
2. Compliments of 10
3. Any subtraction fact Apply properties of operation 1 and 2 less Decompose/Recompose
1. Building up through 10
2. Building down through 10
Fluent = 1) Just knowing some answers, knowing some answers from patterns (n+ 0=n), and knowing some answers from the use of strategies. (Progressions, pg. 18)

==

Notice how they have n+0 =n.

We were told that was too hard even for teachers.  Here it is on the official website of Illinois.

==It gets worse, this is page 2

Repeated Addition/Skip Counting Repeated Counting on
1. Mental arrays.
2. 5 facts and clocks. Apply properties of operation (Commutative, x 1, x 0…..) Connecting addition doubles and x 2 facts. Associative/Distributive Property-Decompose/Recompose
1. Halve than double: 4 x8 = (2×8) + (2 x 8)
2. Double and 1 more group: 3 x 8 = (2×8) + 8)
3. Square and 1more group: 7 x 6 = (6 x 6) + 6
4. One more group than a known fact: 8 x 6 = (5×8) + 8
5. Build off a known fact. Nifty nines- 9’s relationship to 10.
Division
Relationship between multiplication and division. Near Facts: 50/6 is in between 8 x 6 and 9 x6. Properties of operation.
As should be clear, this is not a matter of instilling facts divorced from their meanings, but rather the outcome of a carefully designed learning process that heavily involves the interplay of practice and reasoning. (Progressions, pg. 27)

==

Keith Devlin eat your heart out.

m’*n = m*n + n

The recursive definition of multiplication.

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As should be clear, this is not a matter of instilling facts divorced from their meanings, but rather the outcome of a carefully designed learning process that heavily involves the interplay of practice and reasoning. (Progressions, pg. 27)

But we were told this was impossible for grade K to grade 2.

The Counting on Rules of Arithmetic (CORA) are precisely this.

Search progressions “counting on”

http://commoncoretools.files.wordpress.com/2011/05/ccss_progression_cc_oa_k5_2011_05_302.pdf

“Progressions for the Common Core
State Standards in Mathematics (draft)”

The Common Core Standards Writing Team
29 May 2011
Draft,

“K, Counting and
Cardinality; K–5,
Operations and Algebraic
Thinking”

Grade K onwards.

From page 5

and creating equivalent but easier or known sums (e.g., adding 6 + 7
by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

6+6′ = (6+6)’ = 12′ = 13

Notice how the prime notation for counting on by one makes the logic easier to understand.

How do we deal with 6+7 if we already know 6+6=12?  We write 6+7 as 6+6′ and then we apply

6+6′ = (6+6)’ = 12′ = 13

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http://www.youtube.com/watch?v=W0OfkpzVhcw

TouchMath Kindergarten Software Disc 1 — Addition (Addition with Counting On)

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4th grade commutative video

Associative property videos

http://www.youtube.com/channel/HCFnjDjykrY_Q?feature=relchannel

With the Counting on Rules of Arithmetic, we can prove the associative law.

At e-vendor for 99 cents, although calling it Peano Axioms

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About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
This entry was posted in Counting on Definition of Addition, Counting on Rules of Arithmetic, Kindergarten Counting On, Peano Axioms for Number Line. Bookmark the permalink.

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