Paper #3a Ict in Math

8/10/2019 Paper #3a Ict in Math

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Jerian Dawn P. Pama MA.Ed. Secondary Mathematics (Non-Thesis) - First Year

a.) (Create a hand-outs using Microsoft Word. Use definition/description of any Mathconcept)

1. Smart Art

West Visayas State UniversityHimamaylan City-Extension

Himamaylan City, Negros Occidental

The Real Number System

Complex Numbers = a+bi= (a,b), i= , i2 = -1, i 3= -i,i4 = 1

Real Numbers= ex: 1, 2, -5, 3,

Rational Numbers= Q

Non-Integers Integers=

Natural Numbersex: {1,2,3,4...}

Whole Numbers ex:{0,1,2,3,4...}

Negative Numbers ex:{...,-4,-3,-2,-1}

Even and OddNumbers

Irrational Numbers = Q`

Radicals ex: 2, 3 TranscedentalNumbers ex:

Imaginary Numbers ex:3, i, 4i, 2


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Operation and Grouping Symbol

- The grouping of algebraic symbols and the sequence of arithmetic operations relyon grouping symbols to ensure that the language of algebra is clearly read.

Grouping symbols include:

= parenthesis/parentheses = brackets = braces = vinculum/vincula

The Number Line

- The set of integers consists of the positive integers, negative integers and zero.( -4,-3,-2,-1,0,1,2,3,4)

-4 -3 -2 -1 0 1 2 3 4

Absolute Value

- For any number n| | || Examples:

1. | 6| 6 2. |9| 9 3. |4 7| 4 7


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Order on the Number Line

-4 -3 -2 -1 0 1 2 3 4

Note:1. As you move to the right on the number line, integers get larger in value.

2 . As you move to the left on the number line, integers get smaller in value.

Example: Write a true sentence using > or


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2. 421 5. -65870

3. 2573

C. Find the absolute value of the following.

1. | | 4. |0| + |5| 2. |-8| - |6| 5. |-12| x -|-5|

3. |-31|

D. Write these integers in order from greatest to least.

1. 8, -13, -19, 0, 11, -15 2. -9, 3, -23, 6, -7 3. 0,-1, -3, 6, 5, -4, 7

2. Equation

a. Linear Algebra

Matrix

- A rectangular array of numbers denoted by 11 1 11 1 .Example: 2 5, 3 6 92 4, 4 525 6

Operations on Matrix

1. AdditionLet

[] [] of the same size m n, then A+B=

[] +

for all i= 1,2, , m and j= 1,2, , n.

Example: 2 4 3 5, 34 5 2 find A+B.Solution: 2 43 3 4 5 5 2 = 33 5 7


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2. Scalar MultiplicationLet [] of size m n and r , rA= [ ] where , where i= 1,2, , m and j=1,2, , n.

Example: 5, 2 4 3 5find rA.Solution: 5 2 4 3 5 = 2 55 25

3. MultiplicationLet [] of size m n [] of the same size n p, then the productAB= C =[ ] of size n p defined by = 11 whenk=1.

i = 1,2,m j = 1,2,p

Example: 2 43, 2 34 ,find AB.Solution: 22 4 4 23 4 24 2 3 43 3 3

AB= 2 6 64 3 2

Exercises: Given 2 3, 2 , 23 4 5, 2 3 43 8Find:

1. A+2B 4. CD2. AB 5. DC3. BA

Solutions:

1. A+2B= 2 3 22 = 2 3 4 22= 4 2 2 3 2= 5 4 5 2. AB= 2 3 2 = 2 2 22 3 3= 2 32 2

-Numbers 3, 4 and 5 will serve as their assignment.


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b. Number Theory

Summation and Multiplication Symbol

(Summation) (Multiplication)

- are used to simplify the writings of sum and product

1 1 - Indicates the summation of x sub i, where i ranges from 1 to n is equal

to 2 3 , where i is called index of summation.

Properties of Summation

3 1 1 1 1

2 1 1 4 11 1 1

Special Symbol


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Multiplication Notation

1 1 Certain Preposition

1 1 1 2 1 3 1 1 4 1

Examples: Expand

1 2 2 21

3 5 2 4 411

5 1


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Use the Summation Notation:

1. 11 1 2. ( 1 2 3

11

3. 1 2 + 3 + 1

-

The examples (Expand) will serve as their exercises.


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c. Algebra

Radical Expression

Radical - comes from the Latin word radix which means root.

Parts of the Radical Expression

radical symbol

index radicandDefinitions:

1. Radical Symbol- the symbol which indicates a root of a number.2. Radicand - the number inside the radical sign or the number whose root is beingconsidered.

3. Radical - an expression consisting of a radical sign and a radicand.

Note:

1. If x is not a perfect square, then is an irrational number.2 . If x is a perfect square, then is a rational number.3 . A rational number is a number which can be expressed as a ratio of two integers.Examples;1. 2. 25 5 3.

36 6

4. 2 4 42 5.


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Cube Roots and Other Roots

Example:

, , Definition: y is the nth root of x if yn = x where n must be a natural number greater than1.

yn = x y= Note: In using

1. The number of roots depends upon the index n.2. If the index n is odd , whether x is positive or negative , then x will have a uniqueroot . That is the nth root of x is positive if x is positive and the nth root is negative if x isnegative.

3. If the index is even and x is positive , then x has 2 real nth roots , one is positiveand the other is negative. The positive root is the principal root.

4. If the index is even and x is negative , then x has no real nth root .

Examples:

1. = 2 4. = no solution 2. = -2 5. = 1.5874 3. = -4


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Laws on Radicals

1. ( )n = a, a>0

2 =

3. 4. Examples:

1. 2

=

3. 4. 5.

Activity: Simplify the following radicals.

1. 2. 3. 4. 5.


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d. Calculus- Integral Calculus

Basic Integration Formulas

- From a knowledge of differentiation we obtain the following results which arefundamental and should be memorized.

1. 2. 3. 4. 5. 6. 7.

Examples: Integrate the following.

1. Solution: 2.

Solution:


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3.

Solution:

4. Solution:

5.

Solution:


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Exercises: Integrate the following function.

1.

2. 3. 4. 5.


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e. Trigonometry

Trigonometric Identities

Identity is an equation that satisfies every value of its variable for which eachside of the equation is defined.- is an equation involving a circular function or combination of

circular functions which are valid for all values of the angle for whichthe functions are defined.

The Eight Fundamental Identities

A. Reciprocal Relations

1. sec 2 . csc

3.

B. Quotient Relations

1. 2.

C. Pythagorean Relation1 . 2.

3.


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Examples: Simplify each of the following

1. 2.

Solution:

3. 4. Solution:

5. Proving Identities

- To verify identities, it is necessary that we have a good mastery of the fundamentalidentities. Here are some suggestions that will make the work easier.

1. Memorize the eight fundamental identities.2. Determine the complicated side and start working on the equation try totransform it into the less complicated side.

3. Keep an eye on the other side as you work with the identity it`s easier to hit atarget that you can see.

4. It is much easier to begin by changing all functions into sine and cosinefunctions.

Note: Do not cross multiply the sides of trigonometric identities.


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Example: Prove the following

1. Solution: Left Hand Side (LHS)

2. Solution: Left Hand Side (LHS)

3.

Solution: Right Hand Side (RHS)


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Exercises: Prove the following

1. 2.

3. 4. 5. 6. 7. B


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3. Geometric Figure with labels with Drawing Canvass

Polygons

-

is a union of a finite number of line segments satisfying the following conditions:(i) Each end point is an end point of exactly two segments;(ii) No two segments intersect except at an end point; and(iii) Two segments with a common end point are not collinear.

We specify a polygon by listing its vertices it order, such as ABCD. This polygon iscalled a quadrilateral , meaning it has four sides AB, BC, CD, and DA .

Illustration:

AB

CD


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Specials Types of Quadrilaterals

1. Parallelogram - opposite sides are parallel.

Illustration:

2. Rectangle - its angles are all right angles.

Illustration:

A B

C D

A

B

C

D


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3. Square - a rectangle with two adjacent sides congruent.

Illustration:

4. Rhombus - all its sides are congruent.

Illustration:

C D

BA

D

C

B

A


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5. Trapezoid - just two of its sides are parallel. The parallel sides are called the bases andthe nonparallel sides are called the legs.

Illustration:

Exercises

1. Explain why the quadrilateral ABCD can also be described as BCDA or DCBA, but notACBD.

2. Prove that each rectangle is a parallelogram.3. Prove that opposite sides of a parallelogram are congruent.4. Let M and N be the two midpoints of the legs (nonparallel sides) of a trapezoid. Provethat MN is parallel to each base (parallel side) of the trapezoid.5. Exercise 2 states that all rectangles are parallelograms. It is clear from the definition thatall squares are rectangles. List all such relationships between any two types ofquadrilaterals listed above.6. Are all triangles (3-sided polygons) convex? Are all quadrilaterals (4 sided polygons)convex? Are all pentagons (5-sided polygons) convex?7. The following theorem is very useful to carpenters who want to make sure that thewalls of a room are square. Show that a parallelogram ABCD is a rectangle if and only if AC

= BD.

A B

C D

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Paper #3a Ict in Math

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