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Computational estimation is the process of forming an approximate answer to a numerical problem. This is especially useful when the computation is done on a calculator.
Mental mathematics is the process of producing an answer to a computation without using computational aids. It can help in our everyday estimation skills. It is essential that we have these skills even in a time when calculators are readily available.
Repeated-Subtraction Model
Properties of Multiplication
Closure property of multiplication : For whole numbers a and b, a*b is a unique whole number.
Commutative property: For whole numbers a and b, a*b = b*a.
Associative Property: For whole numbers a,b, and c, (a*b)*c = a*(b*c).
Identity property of multiplication : There is a unique whole number 1 such that for any whole number a, a*1 = a = 1*a.
Zero multiplication property : For any whole number a, a*0 = 0 = 0*a.
Area Model
Cartesian-Product Model
The Cartesian-Product model offers another way to discuss multiplication.
Repeated-Addition Model
When we put equal-sized groups together we can use mulipication. We write 3+3+3+3+3 as 3x5 and say "three times five" or "three multiplied by five." The repeated-addition model can be illustrated in several ways, including number lines and arrays.
An algorithm (named for the ninth-century Arabian mathematician Abu al Khwarizmi) is a systematic procedure used to accomplish an operation.
Properties of Subtraction
Commutative property does not exist for subtraction : Say we have to compute 2 - 3 , now if we do 2-3 = -1 and if we place the number as 3-2, it equals 1. Hence Commutative property is not applicable to subtraction.
Associative property also does not exist for subtraction. Say we have 2-(3-4) = 2-(-1) = 3. Now if we change the order of subtraction, (2-3)-4 = -1-4=-5. Hence associative property also does not exist for subtraction.
Identity : Same as that of Additive identity. 0+(-N) = -N , where N is positive.
Inverse : N is the inverse of a number –N where N is positive, since N-N=0.
Solving Problems by using Model
Missing-Addend Model
Take-away Model
Properties of Addition
Closure Property of Addition : If a and b are whole numbers, then a + b is a whole number.
Commutative Property of Addition : If a and b are any whole numbers, then a + b = b + a.
Associative Property of Addition : If a, b, and c are whole numbers, then (a + b) + c = a + (b + c).
Identity Property of Addition : There is a unique whole number 0, the additive identity, such that for any whole number a, a + 0 = a = 0 + a.
Solving problems by using Model
Set Model
A set model is one way to represent addition of whole numbers.
Distributive Property of Set Intersection over Union
For all sets A, B, and C, A ∩ (B ∪ C)= (A ∩ B) ∪ (A ∩ C) is the distributive property of set intersection over union.
Associative Property of Set Union
A ∪ (B ∪ C) = (A ∪ B) ∪ C is the Associative property of set union.
Associative Property of Set Intersection
The property A ∩ (B ∩ C) = (A ∩ B) ∩ C is the associative property of set intersection.
Set Union
The union of two sets A and B is the set of elements, which are in A or in B or in both. It is denoted by A ∪ B and is read ‘A union B’
Set Interaction
The intersection of two sets X and Y is the set of elements that are common to both set X and set Y. It is denoted by X ∩ Y and is read ‘X intersection Y’.
We use 100 to represent ten 10s, or one 100. In the base-five system, we need a symbol to represent five 5s. The number 100 means 1•10² + 1•10¹ + 1•1, whereas the number 100 base five means 1•5² + 1•5¹ + 1•0, or 25.
Hindu-Arabic
Today, we use Hindu-Arabic numeration system which all numerals are constructed from the 10 digits-0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, and it’s place value is based on powers of 10, the number base of the system.
Roman
Greek
Mayan
Egyptian
Babylonian
A sequence is a set of numbers in a specific order. What this means is that the set of numbers can be put into a one-to-one correspondence with the Counting Numbers (1, 2, 3, 4, ... ).
Figurate numbers provide examples of sequences that are neither arithmetic nor geometric. Such numbers can be represented by dots arranged in the shape of certain geometric figures. The number 1 is the beginning of most patterns involving figurate numbers. Here is a website that shows a great example of Pascal’s Triangles. It also contains references about the triangle numbers.
Not only Pascal’s triangles but also a sequence such as 3, 3, 3, 3, 3,…was interesting that it is considered both an arithmetic and geometric sequence because you can both multiply 1s and add 0s to get 3. I didn’t think this could be a sequence before I took this class, so it was a fresh shock to me.
The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci (a contraction of filius Bonacci, "son of Bonaccio").
A geometric sequence is a sequence such that each successive term is obtained from the previous term by multiplying by a fixed number called a common ratio.
The sequence 5, 10, 20, 40, 80, .... is an example of a geometric sequence. The pattern is that we can multiply by a fixed number of 2 to the previous term to get to the next term. It is geometric if you are always multiplying by the same number each time.
An arithmetic sequence is a sequence in which the difference between any two consecutive terms is the same, i.e., the difference is a constant.
In order to identify if a pattern is an arithmetic sequence you must examine consecutive terms. If all consecutive terms have a common difference you can conclude that the sequence is arithmetic.
Consider the sequence: 3, 10, 17, 24, 31, ...
Since 10 - 3 = 7, 17 - 10 = 7, 24 - 17 = 7, 31 - 24 = 7 the sequence is arithmetic. We can continue to find subsequent terms by adding 7. Thus, the sequence continues : 38, 45, 52, etc.
If a/b and c/d are any rational numbers,
then a/b ÷ c.d = e/f if, and only if, e/f is the unique rational number such that d/c * e/f = a/b.
If a/b and c/d are any rational numbers,
then a/b * c/d = a*c / b*d.
Multiplication with Mixed Numbers
Properties of Multiplication of Rational Numbers
Multiplication Property of Zero for Rational Numbers
Multiplication Property of Zero for Rational Numbers:
If a/b is any rational number,
then a/b * 0 = 0 = 0 * a/b.
Multiplication Property of Inequality for Rational Numbers
Multiplication Property of Inequality for Rational Numbers:
1) If a/b > c/d and e/f, then a/b * e/f > c/d * e/f.
2) If a/b > e/c and e/f < 0, then a/b * e/f < c/d * e/f.
Mutiplication Property of Equality for Rational Numbers
Mutiplication Property of Equality for Rational Numbers:
If a/b and c/d are any rational numbers such that
a/b = c/d, and e/f is any rational number then,
a/b * e/f = c/d * e/f.
Distributive Property of Multiplication Over Addition for Rational Numbers
Distributive Property of Multiplication Over Addition for Rational Numbers:
If a/b, c/d, and e/f are any rational numbers, then
a/b(c/d + e/f) = (a/b * c/d) + (a/b * e/f)
Multiplicative Identity and Multiplicative Inverse of Rational Numbers
Multiplicative Identity and Multiplicative Inverse of Rational Numbers:
1. The number 1 is the unique number such that for every rational number 1/b, 1*(a/b) = a/b = (a/b)*1.
2. For any nonzero rational number a/b, b/a is the unique rational number such that a/b * b/a = 1 = b/a * a/b.
If a/b and c/d are any rational numbers,
then a/b - d/c is the unique rational number e/f
such that a/b = d/c + e/f.
If a/b and c/d are any rational numbers,
then a/b - c/d = a/b + (-c/d)
If a/b and c/d are any rational numbers,
then a/b - c/d = ad-bc/bd.
Additive Inverse Property of Rational Numbers:
For any rational number a/b, there exists a unique rational number -a/b. the additive inverse of a/b,
such that a/b + (-a/b) = 0 = (-a/b) + a/b.
Addition Property of Equality:
If a/b and c/d are any rational numbers such that
a/b = c/d, and if e/f is any rational number,
then a/b + e/f = d/c + e/f.
If a/b and c/b are rational numbers,
then a/b + c/b = a+c/b.
If a/b and c/d are any two rationanl numbers,
then a/b + c/d = ad+bc/bd.
Given two different rational numbers a/b and c/d, there is another rational number between these two numbers.
If a, b, and c are integers and b > 0, then a/b > c/b if, and only if, a>c.
Two fractions a/b and c/d are equal if, and only if,
ad = bc.
A rational number a/b is in simplest form if b > 0 and GCD(a,b) = 1; that is, if a and b have no common factor greater than 1, and b > 0.
Ex)
80/120 = 8*10/12*10 = 8/12
then, 8/12 = 2*4/3*4 = 2/3.
Hence, 80/120 = 2/3.
Fundamental Law of Fractions
Let a/b be any fraction and n a nonzero integer. Then, a/b = an/bn
Sieve of Eratosthenes is a method the Greek mathematician Eratosthenes developed for identifying prime numbers.
A factorization containing only prime numbers is a prime factorization.
If a and b are any integers, then b divides a, written b|a, if, and only if, there is a unique integer q such that a = bq.
Divisibility Test for 6
Divisibility Test for 6
An integer is divisible by 6 if, and only if, the integer is divisible by both 2 and 3.
Divisibility Test for 11
Divisibility Test for 11
An integer is divisible by 11 if, and only if, the sum of the digits in the places that are even powers of 10 minus the sum of the digits in the places that are odd powers of 10 is divisible by 11.
Divisibility Test for 9
Divisibility Test for 9
An integer is divisible by 0 if, and only if, the sum of the digits of the integer is divisible by 9.
Divisibility Test for 3
Divisibility Test for 3
An integer is divisible by 3 if, and only if, the sum of its digits is divisible by 3.
Divisibility Test for 8
Divisibility Test for 8
An integer is divisible by 8, if, and only if, the last three digits of the integer represent a number divisible by 8.
Divisibility Test for 4
Divisibility Test for 4
An integer is divisible by 4 if, and only if, the last two digits of the integer represent a number divisible by 4.
Divisibility Test for 10
Divisibility Test for 10
An integer is divisible by 10, if, and only if, its units digit is divisible by 10; that is, if, and only if, the units digit is 0.
Divisibility Test for 5
Divisibility Test for 5
An integer is divisible by 5, if, and only if, its units digit is divisible by 5; that is, if, and only if, the units digit is 0 or 5.
Divisibility Test for 2
Divisibility Test for 2
An integer is divisible by 2, if, and only if, its units digit is divisible by 2.
For any integers a and b, a is less than b,
written a < b, if, and only if, there exists a positive integer k such that a + k = b.
Theorem
5-10 : a < b (or equivalently, b > a) if, and only if, b - a is equal to a positive integer; that is, b - a is greater than 0.
5-1 1: a) If x < y and n is any integer,
then x + n < y + n.
b) If w < y, then -x > -y.
c) If x < y and n < o, then nx > ny.
d) If x < y and n < 0, then nx < ny.
If a and b are any integers, then a/b is the unique integer c, if it exists, such that a = bc.
Properties of Integer Multiplication
Closure property : ab is a unique integer.
Commutative property : ab = ba.
COmmutative property : (ab)c = a(bc).
Multiplicative properties of multiplication over addition for integers : a(b + c) = ab + ac and (b + c)a = ba + ca.
Multiplicative properties of multiplication over subtraction for integers : a(b - c) = ab - ac and (b - c)a = ba - ca.
Zero multiplication property : 0 is the unique integer such that for all integers a, a*0 = 0 = 0*a.
Chip Model, Number-Line Model
Integer subtraction can be modeled with a charged field. For example, consider -2 - -7. To subtract -7 from -2, we first represent -2 so that at least 7 negative charges are present. To subtract -7, remove the 7 negative charges, leaving 5 positive charges.
Hence, -2 - -7 = 5.
Closure property : a + b is a unique integer.
Commutative property: a + b = b + a.
Associative property : (a + b) + c = a + (b + c).
Identity : 0 is the unique integer such that, for all integers a, 0 + a = a = a + 0.
Distance is always a positive number or zero. The distance between the point corrsponding to an integer and 0 is the absolute value of the integer.
Number-Line Model
Charged-Field Model
A model similar to the chip model uses positive and negative charges. A field has 0 charge if it has the same number of positive (+) and negative (-) charges. It can be represented in many ways using the charged-field model.
Chip Model
A function from set A to set B is a correspondence from A to B in which element of A is paired with one, and only one, element of B.
It is one of the most widely recognized representations of a function is as a graph.
It is another useful way to describe a function. The function could be given using ordered pairs. For example, when 0 is the input and 1 is the output, that is recorded as the ordered pair (0, 1).
It can be used to examine whether a correspondence represents a function.
We can write an equation to depict the rule. The output values can be obtained by substituting the values 0, 1, 3, 4, and 6 for x in f(x) = x + 7, as shown:
f(0) = 0 + 7 = 7
f(1) = 1 + 7 = 8
f(3) = 3 + 7 = 10
f(4) = 4 + 7 = 11
f(6) = 6 + 7 = 13
It is another way to prepare students for the concept of a function.What goes in the machine is referred to as input and what comes out as output.
It is like a game called guess my rule. We figure out a certain rule from the question.
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations.
For instance, in the formula x + 3 = 5, x is a variable which represents an "unknown" number.
An algebraic expression is a mathmatical expression containing variables, numbers, and operation symbols.
