1.1 The counting numbers
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
