# PowerPoint for Chapter 2

```Reconceptualizing
Mathematics
Part 1: Reasoning About Numbers and Quantities
Judith Sowder, Larry Sowder, Susan Nickerson
CHAPTER 2 – NUMERATION
SYSTEMS
1
The need to quantify and express values of
quantities have led to the invention of
numeration systems. A variety of words and
symbols, called numerals, have been used to
communicate number ideas. Our Hindu-Arabic
numeration system uses ten digits, 0–9.
2-2
ACTIVITY
Below are different ways of representing twelve. Can
you deduce what each individual mark represents? How
would the number ten have been written in each case?
2-3
Some ancient cultures did not need many number
words. For example, in a recently discovered culture in
Papau New Guinea, the same word—“doro”—was used
for 2, 3, 4, 19, 20, and 21. By pointing also to different
parts of the body and saying “doro,” these people could
tell which number was intended.
2-4
DISCUSSION
2-5
2.1
2-6
Example:
2-7
In our base ten system, the whole-number
place values result from groups of ten—ten
ones, ten tens, ten hundreds, and so on. This is
because our system works fine until we have ten
of something because there is no single digit
meaning ten. So, for example, when we reach
ten ones, that’s when we name one ten, with
zero ones left over.
2-8
EXAMPLE
2-9
DISCUSSION
2-10
The decimal point indicates that we are breaking up the
unit one into tenths, hundredths, thousandths, and so
on. The number one is the focal point of the system.
For instance, .642 is 642 thousandths of one.
Example:
.6 is six-tenths of one
6 is six-tenths of ten
70 is seven-tenths of one hundred
.007 is seven-tenths of .07
.08 is eighty thousandths
2-11
2.2
2-12
2-13
Suppose we live in “cartoon land,” and instead
of having ten digits, we only have eight (0, 1, 2,
3, 4, 5, 6, 7). Then, counting would look like:
0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16,
17, 20, 21, 22, 23, 24, 25, 26, 27, 30, 31…
To let everyone know what base one is working
with, we would write our numbers in this
system with a subscript of “eight.”
For example, 10eight (we would say “one-zero”)
2-14
ACTIVITY
2-15
EXAMPLE
2-16
DISCUSSION
How could we find a way to write the number
6072 in base eight?
2-17
ACTIVITY
What are the first 20 numbers in base three?
What is 1000three in base ten?
Write 547 in base five.
What are the place values in a base “b” system?
What digits would be needed?
What is 18 in base two?
2-18
BASE BLOCKS
It is helpful to think of other bases using base
blocks. We represent them as follows:
These are labeled as “small cube,” “long,” “flat,” and “large
cube,” respectively.
Base four, for example, would have zero, one, two or three
of any of the given types of blocks.
2-19
DISCUSSION
2-20
2-21
2.3
2-22
2-23
EXAMPLE
Make sure you understand the methods used on both sides here.
2-24
EXAMPLE
2-25
2-26
2-27
ACTIVITY
2-28
EXAMPLE
2-29
EXAMPLE
2-30
ACTIVITY
Subtract 231four from 311four in base four using
only drawings.
2-31
2.4
2-32
Understanding place value is absolutely foundational for
elementary age children. Even so, oftentimes children
lack true understanding of what we even mean by “base
ten”. By working with other bases, hopefully you’ve
come to a deeper understanding of place value.
One activity centered program asks children to begin
grouping by twos, threes, and so on even before
extensive work in the traditional base ten.
2-33
The manner in which we vocalize numbers is also
important. Some young U.S. children will write 81 for
eighteen.
Place value instruction in schools has become a matter
of memorizing. For instance, the 7 in 7200 is known to
be the “thousands place,” but children are not able to
answer the question of how many hundred dollar bills
are in \$7200.
Using base blocks has been a successful way for children
to get a better feel for place value.
2-34
Also, children often don’t understand the true meaning
of decimal places. They don’t understand that, for
instance, .642 is 642 thousandths of one.
Further, that we use tens and tenths, hundreds and
hundredths, causes children to lose sense-making when
it comes to decimals.
When teachers say, instead, “two point one five” for
2.15, it removes any sense for the number itself.
Plan to give additional emphasis when you use the “th”
sound with children.
2-35
DISCUSSION
When comparing .4 and .40 one student stated
that it was necessary to, “...add a zero to the
end of .4 so that the numbers are the same
size.”
What was the misconception behind the student’s
thinking?
Which do you think the student thought was the larger
of the two numbers?
How would you correct this error when working with
such a student?
2-36
If teachers postpone work with operations on
decimals until students conceptually understand
these numbers, students will be much more
successful in the long run.
Research has shown that once students have
learned only rote rules for calculating decimals,
that it is extremely difficult for them to re-learn
how to calculate them meaningfully later on.
2-37
continued….
2-38
2-39
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