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CHAPTER 9: THE MATHEMATICAL WORLD

**A. NUMBERS**
**K-2**
Numbers can be used to count things, place them in
order, or name them.
Sometimes in sharing or measuring there is a need
to use numbers...
It is possible (and often useful) to estimate quantities
without knowing...
Simple graphs can help to tell about observations.
**3-5**

The meaning of numerals in many-digit numbers depends
on their positions.
In some situations, "0" means none of something,
but in others it may...
When people care about what is being counted or measured,
it is...
Measurements are always likely to give slightly different
numbers,...
**6-8**

There have been systems for writing numbers other
than the Arabic...
A number line can be extended on the other side of
zero to represent...
Numbers can be written in different forms, depending
on how they are...
The operations + and - are inverses of each other--one
undoes what the...
The expression a/b can mean different things: a parts
of size 1/b each,...
Numbers can be represented by using sequences of
only two symbols...
Computations (as on calculators) can give more digits
than make sense...
**9-12**

Comparison of numbers of very different size can
be made...
Numbers can be written with bases different from
ten (which people...
When calculations are made with measurements, a small
error in the...
The effects of uncertainties in measurements on a
computed result can...

**B. SYMBOLIC RELATIONSHIPS**
**K-2**
Similar patterns may show up in many places in nature
and in the...
Sometimes changing one thing causes changes in something
else. In...
**3-5**

Mathematical statements using symbols may be true
only when the...
Tables and graphs can show how values of one quantity
are related to...
**6-8**

An equation containing a variable may be true for
just one value of the...
Mathematical statements can be used to describe how
one quantity...
Graphs can show a variety of possible relationships
between two...
**9-12**

In some cases, the more of something there is, the
more rapidly it may...
Symbolic statements can be manipulated by rules of
mathematical...
Any mathematical model, graphic or algebraic, is
limited in how well it...
Tables, graphs, and symbols are alternative ways
of representing data...
When a relationship is represented in symbols, numbers
can be...
The reasonableness of the result of a computation
can be estimated...

**C. SHAPES**
**K-2**
Shapes such as circles, squares, and triangles can
be used to describe...
**3-5**

Length can be thought of as unit lengths joined together,
area as a...
If 0 and 1 are located on a line, any other number
can be depicted as a...
Graphical display of numbers may make it possible
to spot patterns...
Many objects can be described in terms of simple
plane figures and...
Areas of irregular shapes can be found by dividing
them into squares...
Scale drawings show shapes and compare locations
of things very...
**6-8**

Some shapes have special properties: Triangular shapes
tend to make...
Lines can be parallel, perpendicular, or oblique.
Shapes on a sphere like the earth cannot be depicted
on a flat surface...
The graphic display of numbers may help to show patterns
such as...
It takes two numbers to locate a point on a map or
any other flat...
The scale chosen for a graph or drawing makes a big
difference in how...
**9-12**

Distances and angles that are inconvenient to measure
directly can be...
There are formulas for calculating the surface areas
and volumes of...
Geometric shapes and relationships can be described
in terms of...
Different ways to map a curved surface (like the
earth's) onto a flat...

**D. UNCERTAINTY**
**K-2**
Some things are more likely to happen than others.
Some events can be...
Often a person can find out about a group of things
by studying just a...
**3-5**

Some predictions can be based on what is known about
the past,...
Statistical predictions (as for rainy days, accidents)
are typically better...
Summary predictions are usually more accurate for
large collections of...
Spreading data out on a number line helps to see
what the extremes...
A small part of something may be special in some
way and not give an...
**6-8**

How probability is estimated depends on what is known
about the...
Probabilities are ratios and can be expressed as
fractions, percentages,...
The mean, median, and mode tell different things
about the middle of...
Comparison of data from two groups should involve
comparing both...
The larger a well-chosen sample is, the more accurately
it is likely to...
Events can be described in terms of being more or
less likely,...
**9-12**

Even when there are plentiful data, it may not be
obvious what...
When people estimate a statistic, they may also be
able to say how far...
The middle of a data distribution may be misleading--when
the data...
The way data are displayed can make a big difference
in how they are...
Both percentages and actual numbers have to be taken
into account in...
Considering whether two variables are correlated
requires inspecting...
The larger a well-chosen sample of a population is,
the better it...
A physical or mathematical model can be used to estimate
the...

**E. REASONING**
**K-2**
People are more likely to believe your ideas if you
can give good...
**3-5**

One way to make sense of something is to think how
it is like...
Reasoning can be distorted by strong feelings.
**6-8**

Some aspects of reasoning have fairly rigid rules
for what makes sense...
Practical reasoning, such as diagnosing or troubleshooting
almost...
Sometimes people invent a general rule to explain
how something...
People are using incorrect logic when they make a
statement such as...
A single example can never prove that something is
always true, but...
An analogy has some likenesses to but also some differences
from the...
**9-12**

To be convincing, an argument needs to have both
true statements and...
Logic requires a clear distinction among reasons:
A reason may be...
Wherever a general rule comes from, logic can be
used in testing how...
Once a person believes in a general rule, he or she
may be more likely...
Very complex logical arguments can be made from a
lot of small...