Chapter 4 Whole Numbers

Distributive law for division over addition

(a+b)/c=(a/c)+(b/c)

Properties of Whole Number Multiplication

1. Closure Property 2. Communitive Property 3.Associative Property 4. Identity 5. Zero Product 6. Distributive of Multiplication over addition

"Properties" of Division

1. Not communitive 2. not closed 3. not associative 4. no identity 5.Distributive law for division over addition 6. division by zero is undefined

Whole Number Multiplication Approaches

1. Repeated Addition 2. Cartesian Product

Ways to represent Cartesian Product Approach

1. Set Model 2.Tree Diagram

3 Approaches to Subtraction

1. Take Away 2.Comparison 3. Missing Addend

Estimating Addition and Subtracting Methods

1. Truncation 2.Rounding

Ways to represent the missing addend approach

1. sets 2. measurement line

Approaches to whole number division

1.Repeated subtraction 2.Partitioning or Fair Shares 3. Missing Factor

2 methods for modeling addition

1.Set Model 2.Number Line or Measurement Model

Modeling Repeated Subtraction Method of Division

1.Set model 2.Number line

Ways to represent repeated addition

1.set model 2. number line 3.Array/Area

Take Away Approach counting backwards

7-3 3 is the # we count back ******* "seven" ****** "six" ***** "five" **** "four" <---- the answer

Compensation

A method that makes numbers more compatible by splitting one addend into two numbers so that one of them is more compatible with the other addend

Set Model for Addition

Count each set, combine them, then count the total

Mental Addition and Subtracting Strategies

Counting strategies Left to Right Methods Compatible Numbers Breaking Apart Numbers

Standard Algorithm for Addition

Definition: method where the partial sum is an intermediate step and differs in 2 ways: 1. condenses how the partial sum is written 2.regrouping is done while adding rather than at the end

Lattice Method to Written Algorithms of Addition Definition

Definition: we write the sums from single digit facts in pattern, where ones digit goes into the bottom corner and any tens goes into the upper corner, then you add across the diagonal

Partial Sums Written Algorithm Method for Addition Definition

Defintion: add the same place values to each other and then add the num of each place value together Example 2483 +1375 ------- 8 150 700 +3000 --------- 3858

Exponent

For any whole number, a, and any natural number, n, a^n=a(a)(a)(a)... (n factors)

Identity Property of Addition

For any whole number, a, there exists the unique number 0, such that a+0=a and 0+a=a

Less than using whole number addition

For any whole numbers a & b, if and only if there exists a natural number, n, such that a+n=b then it is said that a is _____ b.

Missing Addend Approach to Subtraction Definition

If a & b are any whole numbers then a-b=c if and only if a=b+c for some whole number c

Definition of Subtracting

If a & b are whole numbers where a=n(A) and b=n(B). If B is less than or equal to A then a-b=n(A-B)

Division with Remainders

If a and b are two whole numbers with b not equal to 0 there exists two hole numbers q and r such that a=bq+r 0<=r<b q=quotient r=remainder

Equal Addition for Compatible Numbers

In mental subtraction we make numbers more compatible by equal additions

Cartesian Product Between two sets definition

Let A and B be two sets, the product of set A with set B, denoted AXB, the set of all ordered pairs, (a,b) where A is an element of A and B is an element of B

Repeated Addition Definition

Let a and b be 2 whole numbers with a not equalling zero, then we define axb as axb=b+b+b+b+...(a-times) "b" gets added "a" times

Missing Factor Approach to Division Definition

Let a and b be any whole numbers with b not equal to 0 then a/b=c if and only if there exists a unique whole number "c" such that b(c)=a

Adding and Subtracting Large Numbers Methods

Methods for... 1. Written Algorithms 2. Mental Computations 3. Estimation 4. Calculators

Methods to Subtracting Large Numbers

Subtracting with base ten blocks standard algorithm equal addition algorithms

Take Away Approach

Subtraction Approach 7-3 a set of 7 blocks and from those 7 you group 3 and remove them

Whole Number Addition

Suppose a & b are whole numbers where a=n(A) and b=n(B). If A and B are disjoint, finite sets, then a+b=n(AUB)

Partitioning or Fair Shares Division Definition

The a/b represents the number of objects in eaCh set when the objects of "a" are equally distrubuted among "b" sets

Array or Area Modeling Definition

The product is represented by a rectangular array or table. The 1st factor tells us the number of rows and the second factor tells us the number of columns

Breaking Apart method

Uses expanded notation for mental addition and subtraction

Repeated Subtraction Division Definition

We define a/b = the number of times we can subtract b from a as long as b is not 0

Fact Families/ strip diagrams

_________________ | ? | |------------| | | | | 2 | 5 | |________|_______| 2+5=? 2+5=7

Distributive over addition property of multiplication

a(b+c)=ab+ac

Properties of Exponents

a^m(a^n)=a^m+n (a^m)^n=a^m(n) a^m(b^m)=(ab)^m

Number line or Measurement Model for Addition

count out the first set and then count on the amount of numbers equal to the second set

Number line modeling repeated addition

first factor "a" is the number of times you move "b" spaces on the number line

Communitive Property of Addition

if a & b are any whole numbers then a+b=b+a

Associative Property of Addition

if a, b, & c are ny whole numbers then a+(b+c)=(a+b)+c

Closure Property of Addition

if the sum of any two numbers in the set is an element back in the set

Written Algorithms Defininion

step by step procedures that explains how to compute an operation

Closure Property of Multiplication

the product of any two whole numbers is a unique whole number

Factoring

the use of the distributive property in reverse. means to pull out any common factors in the addends

Equal Additions Algorithms Definition

uses addition to avoid making exchanges (add the same amount to both the minuend and subtrahend)

Counting Strategies Definition

uses counting on, counting back, and skip ocunting, when sums and differences involve multiples of 5s, 10s, or 100s

Rounding

we consider the next lowest place value to determine which value the number is closest to -if its 4 or less round down -if its 5 or higher round up

Truncation

we cut off a number at a certain place value but without any regard to the digits that come after

Comparison Approach of Subtraction

we find the difference by comparing two sets or measures we always ask the question "How much more does this one have over the other?"

Compatible Numbers Definition

we look for two or more numbers that add or subtract to a multiple of 10,100,1000...

Left to Right Method Definition

work from the left to the right or from larger place values to smaller ones