NCTM Expectations Number and Number Operations
Instructional programs from prekindergarten through grade
12 should enable all students to —
In grades 6 –8 all students should—
Understand numbers , ways of representing numbers, relationships among numbers, and number systems
• work flexibly with fractions, decimals, and percents to solve problems;
• compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line;
• develop meaning for percents greater than 100 and less than 1;
• understand and use ratios and proportions to represent quantitative relationships;
• develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation;
• use factors, multiples, prime factorization, and relatively prime numbers to solve problems;
• develop meaning for integers and represent and compare quantities with them.
Understand meanings of operations and how they relate to one another
• understand the meaning and effects of arithmetic operations with fractions, decimals, and integers;
• use the associative and commutative properties of addition and multiplication and the distributive property of multiplication over addition to simplify computations with integers, fractions, and decimals;
• understand and use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems.
Compute fluently and make reasonable estimates
• select appropriate methods and tools for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situation, and apply the selected methods;
• develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use;
• develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results;
• develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.
“Middle-grades students should also work with integers. In lower grades, students may have connected negative integers in appropriate ways to informal knowledge derived from everyday experiences, such as » below-zero winter temperatures or lost yards on football plays. In the middle grades, students should extend these initial understandings of integers. Positive and negative integers should be seen as useful for noting relative changes or values. Students can also appreciate the utility of negative integers when they work with equations whose solution requires them, such as 2 x + 7 = 1.
”
( NCTM Principals and Standards for School Mathematics. NCTM (2000) p. 217)
1

Negative numbers are numbers that are less than zero. They are the opposite of positive numbers.
For example, if a positive number indicates a bank deposit, then a negative number indicates a withdrawal of the same amount. Negative numbers are usually written by writing a negative sign
(also called a minus sign) raised in front of the number they are the opposite of. Thus the opposite of 7 is written −7. When the set of negative numbers is combined with the natural numbers and zero, the result is the set of integer numbers, also called integers , Z (German Zahl , plural Zahlen ).
When you first learned to count, you started with 1, 2, 3 and kept going until you couldn't remember what came next or grew tired of counting. These positive counting numbers (1, 2, 3, 4, ...) are called natural numbers . The ... means the number list continues on infinitely.
Natural Numbers are counting numbers from 1,2,3,4,5,................
N = {1,2,3,4,5,................}
Whole numbers are natural numbers including zero. They are 0,1,2,3,4,5,...............
W = {0,1,2,3,4,5,..............}
W = 0 + N
2

Integers are set containing the positive numbers, 1, 2, 3, 4, ...., and negative numbers,............-3, -2, -1, together with zero.
Whole Numbers together with negative numbers.
Positive Numbers: Positive numbers are, 1,2 ,3 ,4 ,5................. Positive Numbers: {1, 2, 3, . . .}
Negative Numbers: Negative numbers are, ............-3, -2, -1.
Negative integers: { . . . -3, -2, -1}
Zero is neither positive nor negative, but is both.
In other words, Integers are defined as set of whole numbers and their opposites.
Z = {..., -3, -2, -1, 0, 1, 2, 3, .....}
All numbers of the form , where a and b are integers (but b cannot be zero)
Rational numbers include fractions:
* Fractions less than 1 eg: 1/2 or 3/4
* Fraction greater than 1 eg: 5/2
* Mixed Numbers: 2 1/2 = 5/2
Powers and square roots may be rational numbers if their standard form is a rational number.
In rational numbers the denominator cannot be zero
Example:
2 can be expressed in the form of p/q as 2/1
-13/9 = -1.444.......
4/3 = ±1.333...
1/2 = 0. 5 Rational (terminates)
2/3 = 0.6666666.......Rational (repeats)
5/11 = 0.454545......Rational (repeats)
{ the complement of Q }
Cannot be expressed as a ratio of integers.
As decimals they never repeat or terminate (rational numbers always do one or the other)
They go on for ever or infinity.
Example: square root of 2 = = 1. 41421356......Irrational (never repeats or terminates) pi = p = 22/7 = 3.14159265....... Irrational (never repeats or terminates)
Square root of any number that is not a perfect square is irrational.
Cube root of any number that is not a perfect cube is irrational.
Real Numbers are every number, irrational or rational.
Any number that you can find on the number line.
It is a number required to label any point on the number line; or it is a number that names the distance of any point from 0.
R = Q + Q1
Natural Numbers are Whole Numbers, which are Integers, which are Rational Numbers, which are Real
Numbers.
Irrational Numbers are Real Numbers, but not all Real Numbers are Irrational Numbers.
3

Let’s discuss the answers to the following T/F questions.
1.
True or False: An integer is a rational number.
2.
True or False: A rational is an integer.
3.
True or False: A number is either a rational or an irrational, but not both.
Classify according to number type; some numbers may be of more than one type.
4.
0.45
5.
3.14159265358979323846264338327950288419716939937510...
6.
3.14159
7.
10
8.
5 /
3
9.
1 2 /
3
10.
–sqrt(81)
11.
– 9
/
3
Fill in the following table on your own. Each number is considered a member of which set(s).
7
0.45
3.1415926535...................
3.14159
0
List the type of number: natural through real
5/3
1 2/3 = 5/3
= 1. 41421356......
-9/3 = -3
9/3 = 3
-3/4
Pi π = 3.1428571...
3.144444....... .
4

7
0.45 natural whole integer rational real rational real
3.1415926535................... irrational, real
3.14159
0 rational, real whole, integer, rational, real
5/3
1 2/3 = 5/3 rational, real rational, real
= 1. 41421356...... irrational, real irrational, real
-9/3 = -3 integer, rational, real
9/3 = 3
-3/4
Pi π = 3.1428571...
3.144444....... natural, whole, integer, rational, real natural, whole, integer, rational, real rational, real irrational, real rational, real (since it is a repeating decimal)
True or False: An integer is a rational number.
Since any integer can be formatted as a fraction by putting it over 1 , then this is true.
True or False: A rational is an integer.
Not necessarily;
4 /
1
is an integer, but
2 /
3
is not! So this is false.
True or False: A number is either a rational or an irrational, but not both.
True!
In decimal form, a number is either non-terminating and non-repeating (so it's an irrational) or not (so it's a rational); there is no overlap between these two number types!
Classify according to number type; some numbers may be of more than one type.
0.45
This is a terminating decimal, written as a fraction:
45
/
100
=
9
/
20
. Since this fraction does not reduce to a whole number, then it's not an integer or a natural. And everything is a real, so the answer is: rational, real
3.14159265358979323846264338327950288419716939937510...
You probably recognize this as being pi, though more decimal places than you customarily use. The point, however, is that the decimal does not repeat, so pi is an irrational. irrational, real
3.14159
Don't let this fool you! Yes, you often use something like this as an approximation of pi, but it isn't pi!
This is a rounded decimal approximation, and, since this approximation terminates , this is actually a rational, unlike pi which is irrational! The answer is: rational, real
10 Obviously, this is a counting number. That means it is also a whole number and an integer. Depending on the text (there is some inconsistency 10/1), this may also be counted as a rational, which technically-speaking it is. And of course it's also a real. The answer is: natural, whole, integer, rational (possibly) , real
5 /
3
This is a fraction, so it's a rational. It's also a real, so the answer is: rational, real
1 2 /
3
This can also be written as
5 /
3
, (same as the previous problem.) The answer is: rational, real
–sqrt(81)
Your first impulse may be to say that this is irrational, because it's a square root, but notice that this square root simplifies:
– sqrt (81) = –9 , which is just an integer. The answer is: integer, rational, real
– 9
/
3
This is a fraction, but notice that it reduces to
–3
, so this may also count as an integer. The answer is: integer (possibly) , rational, real
5

Too often students explore integers in a rule oriented environment and are given “gimmicks” to perform operations on integers. The first goal of this session is to help teachers explore a more in-depth understanding of addition, subtraction, multiplication, and division, and then to apply that understanding to integers. The second goal is to examine effective teaching strategies used to improve the teaching and learning of integers. A third goal will have participants discuss the pros and cons of more familiar strategies used to teach integers in order for teachers to better understand the importance of connections in meaning making related to integers.
Readiness for integers:
Sample real world uses of directional or signed numbers: temperature football gains and losses scores on card games “in the hole” above and below sea level right and left of zero north and south or east and west of a reference point credit and debit stock market gains and losses above and below ground level (length of roots & stem grown in class)
Familiarity with both vertical and horizontal number lines. Ask questions such as: Why do we have arrows on both ends? Does the 0 need to be in the middle of your drawn line? How do we indicate numbers to the left and numbers to the right? ( raised + and - signs)
Real world uses of integers should help students understand that numbers increase in value as they go to the right or up, and decrease in value as they go to the left or down.
Practice moving a given amount (either positive or negative) from the origin and then from any position.
Guide students to discover the meaning of an integer and its “opposite” integer. Zero is its own opposite. There is symmetry around the zero. How does additive inverse fit into the concept of opposites? Does the property of additive inverse have anything to do with the additive identity element?
Explain the meaning and symbols for absolute value. The absolute value refers only to the magnitude of the number. In mathematics, the absolute value (or modulus) of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3. The absolute value of a number a is denoted by | a |.
6

– 3 + + 4 = + 1
– 3 + – 4 = – 7
– 4 + + 3 = – 1
+ 4 + + 3 = + 7
+ 4 + – 3= + 1
– 4 + – 3 = – 7
Addition of Integers
What is addition? Explore the idea of part -part- whole. When we put 2 or more parts together we get a whole. Mathematically, addend + addend = sum.
Addend + Addend = Sum OR Part + Part = Whole
Do several random examples of simple addition problem such as:
+ 5 + + 4 =
+ 4 + – 2 =
Now do a series of problems such as:
+
+
3 + – 4 = – 1
3 + + 4 = + 7
7

Why is it that sometimes it looks like we are adding the magnitudes and other times it appears we are subtracting? What is similar about the 8 problems? What is different about the problems? Think about the number line, envision the movements then discuss again what is similar about the 8 problems.
Discuss the idea of Addend + Addend = Sum and Part + Part = Whole again as it relates to the numberline difference between model.
Subtraction of Integers
Discuss how addition and subtraction are related. Rewrite an addition problem both horizontally and vertically as a subtraction problem. Which numbers are the parts and which is the whole or total. Also relate this to addend, addend, sum.
3
+ 4
7 – 3 = 4 or 7 – 4 = 3
7
P
+P
W
W – P = P
In Addition we are looking for the total, whole, or sum.
In Subtraction we are looking for the missing part, missing addend, or the difference between the given part and the whole.
Since the children usually understand addition on the numberline, have them do a subtraction problem using the same idea. Addition and subtraction are related, so lay off the addend that is given in the subtraction problem. Ask yourself what is the second addend? If you know the total then how can you find the missing addend on the numberline? Students discover it is the difference between the given addend and the total or sum. The sign of the answer is the direct you go
FROM the given addend TO the sum (You just found the missing addend or the answer to the subtraction problem.)
Just as with the addition problems, investigate a family of subtraction problems and discuss the results. Look for things that are similar and things that are different. From this students have been able to formulate their own rules for solving subtraction of integers.
8

+ 3 – – 4 =
+ 3 – + 4 =
–- 3 – + 4 =
– 3 – – 4 =
+ 4 – – 3 =
+ 4 – + 3=
– 4 – + 3 =
– 4 – – 3 =
Try these on an imaginary numberline.
+ 40 – – 30 =
+ 75 – + 25 =
– 250 – + 300 =
– 125 – – 200 =
Practice on word problems. Make one up using real world applications.
9

How can we mathematically explain that a negative number multiplied by a negative number is a positive number.
Imagine a number line on which you walk. Multiplying xy is taking x steps, each of size y. Think of how many sets of steps in the positive or negative direction need to be taken. So, (x ) (-y) means to stand on zero, face in the negative direction since you are going X sets of y steps in the negative direction. The problem comes when you have a negative number times a negative number. How do you explain a negative set?
+ 2 X + 3 =
+ 2 X – 3 =
– 2 X + 3 =
– 2 X – 3 =
Number of Sets
10

Fill in the table using patterns.
11

:
Imagine a hot air balloon is positioned on top of a mountain. It can rise with helium balloons or fall with sandbags. Remember also that multiplication is modeled by a number of sets times the number in each set:
Let’s model -2 X -3 =
First factor says to either put on (+) or take off (-) the number of sets indicated. Therefore -2 X -3 = would start out “Take off 2 sets of “
The second factor says how many helium balloons (+) or sand bags (-) are in each set.
Therefore, -2 X -3 would be: “Take off 2 sets of 3 sandbags”.
If we can agree that a negative number is just a positive number multiplied by -1, then we can always write the product of two negative numbers this way:
(-a)(-b) = (-1)(a)(-1)(b) = (-1)(-1)ab
For example,
(-2) (-3) = (-1)(2)(-1)(3)
= (-1)(-1)(2)(3)
= (-1)(-1) (6)
So the real question is,
(-1)(-1) = ? and the answer is that the following convention has been adopted:
(-1)(-1) = +1
This convention has been adopted for the simple reason that any other convention would cause something to break.
For example, if we adopted the convention that (-1)(-1) = -1 , the distributive property of multiplication wouldn't work for negative numbers:
(-1)(1 + -1) = (-1)(1) + (-1)(-1)
(-1)(0) = -1 + -1
0 = -2
As Sherlock Holmes observed, "When you have excluded the impossible, whatever remains, however improbable, must be the truth."
Since everything except +1 can be excluded as impossible, it follows that, however improbable it seems, (-1)(-1) = +1.
12

The abstract concept of negative numbers was recognized as early as 100 BC - 50 BC. The
Chinese ”Nine Chapters on the Mathematical Art” (Jiu-zhang Suanshu) contains methods for finding the areas of figures; red rods were used to denote positive coefficients, black for negative.
This is the earliest known mention of negative numbers in the East; the first reference in a western work was in the 3rd century in Greece. Diophantus referred to the equation equivalent to 4x + 20 = 0
(the solution would be negative) in Arithmetica, saying that the equation gave an absurd result.
During the 600s, negative numbers were in use in India to represent debts. Diophantus’ previous reference was discussed more explicitly by Indian mathematician Brahmagupta, in
Brahma-Sphuta-Siddhanta 628, who used negative numbers to produce the general form quadratic formula that remains in use today. However, in the 12th century in India, Bhaskara gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots."
European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debits (chapter 13 of Liber Abaci, 1202) and later as losses (in Flos). At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most nonzero digit of the corresponding positive number's numeral [citation needed]. The first use of negative numbers in a European work was by Chuquet during the 15th century. He used them as exponents, but referred to them as “absurd numbers”.
As recently as the 18th century, the Swiss mathematician Leonhard Euler believed that negative numbers were greater than infinity[citation needed], and it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless, just as René
Descartes did with negative solutions in a cartesian coordinate system. Carl Friedrich Gauss
(1777-1855), one of the most influential scholars in mathematics, enrolled at Göttingen University in Germany in 1795. While Gauss was an undergraduate, he spent three years writing a classic book on the mathematics of integers, Arithmetical Disquisitions . Albert Einstein later praised the importance of Gauss's many contributions to the field of mathematics and showed how Gauss's work helped Einstein develop the theory of relativity. Other European mathematicians were intrigued with the challenge of defining integers. This was a problem that had to be resolved as part of the process of developing definitions of rational arithmetic. The mathematician Leopold Kronecker
(1823-1891) tried to devise a system of math that depended on positive whole numbers and was so convinced of the key role of integers that he declared, only partly in jest, "God made the integers, all the rest is the work of man."
Bourbaki, Nicolas (1998). Elements of the History of Mathematics. Berlin, Heidelberg, and New
York: Springer-Verlag. ISBN 3540647678.
Struik, Dirk J. (1987). A Concise History of Mathematics. New York: Dover Publications.
Werner, M. (1974) Teaching the set of integers to elementary school children . Kendall/Hunt
Publishing Co. Dubuque, Iowa.
Wheeler, E. & Barnard, T. Mathematics Activities for Teaching . Balloon Model.
13