1.1 The counting numbers

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list of cn and the symblols we use to represent the numbers. what are the symbols for the first 9 cn

1 2 3 4 5 6 7 8 9 passed down thru generations tally marks single symbol that represents a group of tally marks (roman numeral 5)

what are counting numbers

1, 2, 3 , 4, 5, 6 etc. not including 0

describe base ten units

10 ones grouped to make a new unit of ten 10 tens grouped to make a new unit of one hundred 10 hundreds grouped to make a new unit of one thousand 10 thousands grouped to make a new unit of ten-thousand group 10 of the previous made units increasingly large units are represented in successive places to the left in a number, so that the value of each place in a number is ten times the value of the place to its immediate right

explain the abstraction of cardinality

3 quantifies a limitless variety of collections. the number 3 is abstract

how does place value require the use of 0

300, 3 in hundreds place, o tens, 0 ones

what is a base ten representation

a string of digits that represents a number, such as 1234, also called base ten expansion

what are two different ways of thinking about the counting numbers

as a list describing a set size

what is a scale

choice of a unit

what are whole numbers

cn together w/ zero 0 1 2 3 4 5 ....

how do we determine how many objects are in a set, and what does this connect to

count the objects in a set list view of the counting numbers w/ the cardinality view

how does place value work

creating larger and larger units by repeatedly bundling them in groups of ten

whats a unit

distance from 0 to 1

once choices for the locations of 0 and 1 have been made,...

each cn is rep by the pt on the number line that is that many units to the right of 0

from 20 onward

english spoken words for cn fit with base ten rep

what is subitizing

for sets of up to about 3, 4 or 5 objects, recognize number of objects in the set immediately, w/o counting one by one immediate recognition of the exact number of objects in a set

what kinds of numbers are there

fractions, decimals, negative , counting, whole numbers

what is a number line

line on which one location has been chose as 0 and another location to the right of 0 as 1 stretch infintiely far in both directioins

what else does the list view and cardinality view have a connection to

numbers later in the list = larger quantities numbers earlier in the list = smaller quantities starting at any cn, the next number in the list describes the size of a set with one more object in it, and the previous number in the list describes the size of a set that has one less object in it

What are numbers and why do we have them

numbers provide us with a means for describing, representing, reasoning about quantities tell us how many or how much to communicate info about a collections of things/quantities of stuff

how do the numbers 10-19 represent the base ten system and place value concept

one group of ten and some ones

what is a number path

path labled with successive cn informal precursors to the concepts of distance/length/number line

what is place value

quantitiy that a digit in a number represents on position of the digit in the number

what is the key innovation of the base 10 system

rather than using new symbols to represent larger and larger numbers, it uses place value

how do you represent zero when respresenting sets of objects

show no objects

what is the expanded form

shows how a number is composed of its base ten units

cn from 11-19 in english does not fit well with their base ten rep

some of the numbers sound like the reverse of one ten and three, one ten and four, one ten and nine (13, 14, 190

what is the list of counting numbers

starts with 1, every number in the list has a unique successor except for the number 1, every number has a predecessor ordered list: every counting number appears exactly once in this list

what is the second way of thinking about the counting numbers

telling how many describes how many things are in a set of things

starting with our modern day symbols how can one continue the list w/o creating an endless string of new symbols

the base ten system decimal system

list view and counting numbers w/ cardinality connection also relies on

the last number we say when we count a set of objects tells us the total number of objects in the set order of the cn is important, unlike order of letters of the alphabet

what is cardinality of a set

the number of things in a set

what is important to consider when considering cardinality of a set

the ordering of the list of counting numbers

distinguish between number lines and number paths

to interpret a number path, number path shows distinct steps to interpret a number line, rely on length/distance from 0 children focus on tick marks instead of length = omitting 0/misinterpretations abt locations of fractions on number lines

what is the base ten system

uses only ten distinct symbols the digits 1-9

how do children extend their understanding of the base ten system

view a group of ten as a unit in its own right understand that a two digit numbers stands for groups of ten and some ones represented with bundled objects/simple drawings

what's a one to one correspondence

when we count a set of objects one by one, we make a one to one correspondence b/w an initial portion of the list of counting numbers and the set ex: when a child makes a correspondence b/w the list 1 2 3 4 5 and the set of blocks. each block is paired with exactly one number and each number is paired with exactly one block

what are the roman symbols

x for 10 l for 50 c for 100 m for 1000


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