Weebly
|MA5.3-5NA : Algebraic Techniques | Mathematics Stage 5 Year 9/10 |
|Summary of Sub Strands |Duration |
|S4 Algebraic Techniques 1 & 2 |5 weeks |
| |Detail: 5 weeks, ….lessons per week (…hours) |
|Unit overview |Outcomes |Big Ideas/Guiding Questions |
|Uses the algebraic symbol system to simplify, |Mathematics K-10 | |
|expand and factorise simple algebraic |MA5.2-6NA simplifies algebraic fractions, and | |
|expressions |expands and factorises quadratic expressions | |
|Simplifies, expands and factorises algebraic |MA5.3-5NA selects and applies appropriate algebraic| |
|expressions involving fractions and negative |techniques to operate with algebraic expressions | |
|and fractional indices | | |
| | | |
|Uses algebraic techniques to simplify | | |
|expressions, expand binomial products and | | |
|factorise quadratic expressions | | |
| | |Key Words |
| | |Algebra, algebraic statement, algebraic symbol, algebraic expression, concrete material, decreasing, |
| | |diagram, equivalent, expand, expression, factorise, grid, increasing letter, multiplication, |
| | |multiplication sign, negative, negative number, number pattern, number plane, number sequence, |
| | |operation, pattern, position number, rule, simplify, symbol, term, value, variable. Algebraic |
| | |fraction, base coefficient, consecutive, distributive Law, expand, exponential notationevaluate, |
| | |factorise, fractional / negative / zero index, highest common factor, like terms, lowest common |
| | |denominator / multiple, power, pronumeral, reciprocal, simplify, square / cube root, |
| | |substituteExpand, Binomial, Quadratic, simultaneous, factorising, etc. |
|Catholic Perspectives |School Free Design |
|TALK TO MICHAEL FOR WHAT TO PUT HERE |This is a free design area for schools to add local additional areas. This could include: |
|CEO will provide guidance in this area |Context if you prefer the unit overview and context to be separate |
|Example: |School focus for learning – eg blooms taxonomy, solo taxonomy, contemporary learning, habits of mind,|
|The value of sacramentality celebrates the presence of God in every facet of creation. |BLP (building learning power) |
|The Christian message is ultimately one of hope |Any specific social and emotional learning which could be embedded into the unit eg enhanced group |
| |work |
|Mathematics, Reality, and God | |
|Paul A. Schweitzer | |
|DOI:10.1093/acprof:oso/9780199795307.003.0013 | |
|Simplicity and symmetry are the heart of beauty in mathematics. Beauty often motivates mathematicians| |
|and physicists. Einstein said that his theory of general relativity had to be true because it was so | |
|elegant. Archimedes was thrilled with his discovery that the ratio of the volume of a cylinder | |
|tightly enclosing the volume of a sphere is 3:2. Mathematics offers beauty without defects. Salvador | |
|Dali produced two religious paintings that have important mathematical components. Mathematics have | |
|very precise norms for proving theorems, but these generally don’t apply to ordinary life or other | |
|academic disciplines. Kurt Gödel brilliantly proved that a mathematical system could be proven either| |
|complete or consistent, but not both. This means mathematics is open to the transcendent, as must | |
|other disciplines be as well, since they are less precise than mathematics. Every type of rational | |
|discourse must be judged according to its own procedures and limitations. By developing n-space, the | |
|mind shows it is made in the image of God. It is helpful to compare theology with mathematics. Both | |
|subjects always have new problems to solve. It is now known that Gödel developed a proof for the | |
|existence of God based on the ontological argument. | |
|Keywords: beauty, golden,mean, Einstein, Dali, Gödel, completeness, consistency, theology, ontologi| |
|cal argument | |
| | |
|Below Connected Website | |
| | |
|Numeracy and the Catholic World View | |
|Numeracy operates within a variety of social contexts. From a Catholic perspective, numeracy must be | |
|imbued with a vision of the innate dignity of all students, as created in the image and likeness of a| |
|loving, generous and creating God. Teachers in Catholic schools have an obligation to not only teach | |
|their students the skills and knowledge to be numerate, but to teach from a Catholic perspective. | |
|Teachers are called to challenge their students to use the skills and knowledge they have acquired to| |
|bring about social change in the world. | |
| | |
|Below is from St Josephs Narrabeen | |
|[pic] | |
| | |
|Below is from Mount St Patrick College, Murwillumbah | |
|PRIMARY AIM | |
|The primary aim of the Department, as a whole, is to inculcate the skills, knowledge and attitudes as| |
|outlined in the syllabuses with a Catholic Perspective. | |
| | |
|GENERAL AIMS | |
|• To provide a structured and caring environment for the learning of Mathematics. | |
|• To develop the significance and relevance of Mathematics in everyday life. | |
|• To attempt to equip all student with the Mathematical skills and knowledge which will help | |
|them to cope with everyday life. | |
|• To make Mathematics meaningful and relevant to students. | |
|• To make Mathematics interesting and enjoyable. | |
|• To teach students to think clearly and logically. | |
|• To teach students good study habits. | |
|• To bring student to the realisation that they are not just learning Mathematics to pass | |
|examinations. | |
|• To develop staff professionally. | |
|• To foster the language of Mathematics as a form of communication. | |
|• To provide a sense of justice and equity in Mathematics regardless of racial origin or | |
|religion and to avoid stereotyping of roles for each sex. | |
| | |
|THE NATURE OF MATHEMATICS LEARNING | |
|Mathematics is learnt by individual students at different rates. | |
|It must be remembered that:- | |
|• students learn best when motivated | |
|• students learn Mathematics through interacting and reflecting. | |
|• students learn Mathematics through investigating. | |
|• students learn Mathematics through language. | |
|• students learn Mathematics as individuals in the context of cultural, intellectual, | |
|physical and social growth. | |
| | |
|CATHOLIC PERSPECTIVE | |
|'We are committed to the development of Catholic schools which demonstrate a special concern for, and| |
|understanding of, the uniqueness of each person.' | |
| | |
|Tick Points History | |
|CATHOLIC (GOSPEL) VALUES: | |
| GV1 Celebration | |
| GV2 Common Good | |
|GV3 Community | |
|GV4 Compassion | |
| GV5 Cultural Critique | |
| GV6 Faith | |
| GV7 Hope | |
|GV8 Human Rights | |
| GV9 Joy | |
|GV10 Justice | |
| GV11 Peace | |
| GV12 Reconciliation | |
| GV13 Sacredness of Life | |
|GV14 Stewardship of Creation | |
| GV15 Service | |
|GV16 Wisdom | |
| | |
|Tick Points Science | |
|Catholic Perspective Keywords | |
|Awe and Wonder | |
|Celebration | |
|Common Good | |
|Charity | |
|Commitment to community | |
|Community Conservation | |
|Compassion | |
|Courage | |
|Cultural Critique | |
|Dignity of each human person | |
|Endurance / perseverance | |
|Faith | |
|Family | |
|Forgiveness | |
|Global Solidarity and the Earth Community | |
|Hope | |
|Hospitality | |
|Human Rights Justice | |
|Joy | |
|Justice | |
|Love | |
|Multicultural Understanding | |
|Peace | |
|Reconciliation | |
|Reverence | |
|Sacredness of Life | |
|Service | |
|Sense of wonder | |
|Servant leadership | |
|Stewardship of Creation | |
|Structural Change | |
|Self Respect (Self Esteem) | |
|Truth | |
|Assessment Overview |
|Critical Question 2: How will we know that students have learned it? |
|As the syllabus outcomes form the focus of the unit, it is necessary to identify the specific evidence of learning to be gathered through teaching, learning and assessment activities that will |
|demonstrate knowledge, skills and understanding. The evidence of learning provides the basis for the selection of content and the planning of the learning experiences within the units. This evidence |
|will enable teachers to make judgements about student achievement in relation to the syllabus outcomes and identified content. |
|Include assessment for, as, of learning |
|Generally, teachers should design specific assessment tasks that can be drawn from a variety of the following sources of information and assessment strategies: |
|• student responses to questions, including open ended questions, |
|• student explanation and demonstration to others, |
|• questions posed by students, |
|• samples of student work, |
|• student produced overviews or summaries of topics, |
|• investigations or projects, |
|• students oral and written report |
|• practical tasks and assignments, |
|• short quizzes |
|• pen and paper tests, including multiple choice, short answer questions and questions requiring longer responses, including interdependent questions ( where one answer depends on the answer obtained|
|in the preceding part) |
|• open book tests |
|• comprehension and interpretation exercise |
|• student produced worked samples, |
|• teacher/student discussion or interviews |
|• observation of students during learning activities including the student’s correct use of terminology |
|• observation of a student participating in a group activity |
|References can be made to the relevant end of chapter review or screening tests found in textbooks or other resource areas |
|Content |Teaching, learning and assessment |Resources |
|Critical Question 1: What should students know and be able to do? |Critical Question 3: How will we structure learning experiences to ensure students | |
|List outcomes and indicators |learn? | |
|This is another opportunity to be explicit about the specific |The learning is planned with identified results and appropriate evidence of | |
|Catholic perspective(s) that students should more fully know and be|understanding in mind. What will be taught (curriculum), and how should it be taught | |
|able to apply, as a result of their engagement in this unit |best (pedagogy), in light of the established goals? What sequence best suits the | |
| |desired results? How will we make learning engaging and effective, given the goals | |
| |and evidence required? | |
| | | |
| |Critical Question 4: How will we respond when students do not learn it or when they | |
| |already know it? | |
| |How do we ensure individual students who need additional time and support for | |
| |learning receive timely and effective intervention? | |
| |How do we differentiate the learning? | |
| |Changes to grouping/instruction | |
| |Different ways to deliver the content | |
| |Different ways for students to demonstrate the learning | |
| |Learning environment is considered | |
| |How will we make learning challenging when student know more than anticipated? | |
| |Adjustments | |
| |Teachers may make adjustments to teaching, learning and assessment practices for some| |
| |students with special education needs, so that they are able to demonstrate what they| |
| |know and can do in relation to syllabus outcomes/catholic perspectives and content. | |
| |The types of adjustments made will vary based on the needs of individual students and| |
| |occurs at the time of learning. | |
| |These may be: | |
| |Adjustments to the assessment process, e.g. additional time, rest breaks, quieter | |
| |conditions, or the use of a reader and /or scribe or specific technology | |
| |Adjustments to assessment activities, e.g. rephrasing questions or using simplified | |
| |language, fewer questions or alternative formats or questions | |
| |Alternative formats for responses, e.g. written point form or notes, scaffolded | |
| |structured responses, short objective questions or multimedia presentations. | |
| | | |
| |Student Reflection | |
| |Students reflect on the demands of the unit of work and the assessment activity. | |
| |They can record their findings about their own processes of learning by constructing | |
| |a PMI chart (plus, minus and interesting) to evaluate the topic | |
| |and the learning by addressing the following questions: | |
| |What did you get the most out of (P)? | |
| |What did you like the best (P)? | |
| |What did you think needed to be developed further (M)? | |
| |What was the most interesting thing you did or learnt (I)? | |
| |How has this unit developed your understanding of the subject? | |
| |What have you learnt about yourself as a learner? | |
| | | |
| | | |
|Stage 5.2 - Algebraic Techniques |Fractions Pre Thoughts from Stage 4 |Math In Living Color ~ Instructions and Questions|
|Students: | |
|Apply the four operations to simple algebraic fractions with |Simplifying Expressions Pre Thoughts From Stage 4 |20Living%20Color/math_in_living_c_o_l_o_r.htm |
|numerical denominators (ACMNA232) | | |
|simplify expressions that involve algebraic fractions with |HCF for Integers and Pronumerals Pre Thoughts From Stage 4 | |
|numerical denominators, | | |
|eg [pic] [pic] [pic] [pic] |Addition and Subtraction | |
|connect the processes for simplifying expressions involving |[pic] | |
|algebraic fractions with the corresponding processes involving |[pic] | |
|numerical fractions (Communicating, Reasoning) [pic] | | |
| | | |
| | | |
| | | |
|Apply the four operations to algebraic fractions with pronumerals |Cancelling of factors within fractions can only occur after factorisation has been |
|in the denominator |done first. |eachingresources/discipline/maths/continuum/Pages|
|simplify algebraic fractions, including those involving |Factorising must happen to both numerator and denominator. |/fracalgebra45.aspx Fractions for algebra and |
|indices, eg [pic] [pic] [pic] |As long as the students realise that they have to factorise first they will be OK |arithmetic |
|explain the difference between expressions such as [pic] and [pic] |Students who don't remember to do this will fall into the trap of trying to cancel |
|(Communicating) [pic] [pic] |anything |pansions_algbrc_fracs.html TIMES Module 25: |
|simplify expressions that involve algebraic fractions, including |Cancelling rules |Number and Algebra: special expansions and |
|algebraic fractions that involve pronumerals in the denominator |% can only cancel numbers through HFC |algebraic fractions - teacher guide |
|and/or indices, |% can only cancel negative symbols if available for both numerator and denominator | |
|eg [pic] [pic] [pic] [pic] |% can only cancel terms of exact consistency to power availability (SEPARATE TERMS AT|Math In Living Color ~ Instructions and Questions|
| |A TIME). 1 for 1 |
| |% can only cancel completely exact SAME brackets |20Living%20Color/math_in_living_c_o_l_o_r.htm |
| |[pic] | |
| |Tidy up Simplifying must be done to finish the solution as its final expression. | |
| |May have improper fractions as result. Mixed numerals don't occur in solutions. | |
|Apply the distributive law to the expansion of algebraic |Pre Thoughts Expanding via Methods of CLAW or BOXED |Distributive Law Interactive CutThe Knot (JAVA |
|expressions, including binomials, and collect like terms where | |Needed) |
|appropriate (ACMNA213) |# EXPANDING and then SIMPLIFYING |
|expand algebraic expressions, including those involving terms with |This can be a combination of expressions - some expanding to do OR many expansions to|/DistributiveLaw.shtml |
|indices and/or negative coefficients, eg [pic] |consider - at then needs to be considered for simplifying via addition/subtraction | |
|expand algebraic expressions by removing grouping symbols and |skills. |Worksheet ~ Distributive Property |
|collecting like terms where applicable, eg expand and simplify |Each question needs to be split up - using the closed bracket as the boundary for |
|[pic] [pic] |separation. |rksheets/Distributive%20Property.pdf |
| |Eg | |
| |[pic] |Many Levels EASY to HARD Ditributive and Simplify|
| | |
| |Each separated expansion is done then results are put together to simplify yet again.|operty |
| |Check expansions and factorisations by performing the reverse process (Reasoning) | |
| |Interpret statements involving algebraic symbols in other contexts eg spreadsheets | |
| |(Communicating) |
| |Explain why an algebraic expansion or factorisation is incorrect eg Why is the |pansions_algbrc_fracs.html TIMES Module 25: |
| |following incorrect? |Number and Algebra: special expansions and |
| |[pic] |algebraic fractions - teacher guide |
|Factorise algebraic expressions by taking out a common algebraic |HCF Pre Test |Factorising By Grouping Questions |
|factor (ACMNA230) |[pic] |
|factorise algebraic expressions, including those involving indices,|- Numerical HCF's that suits EVERY SECTION IN THE EXPRESSION |s&topic_id=73&parent_name=Algebra%26nbsp%3B-%3E%2|
|by determining common factors, eg factorise [pic] [pic] [pic] |[pic] |6nbsp%3BFactoring+-%3E+Grouping |
|[pic] |and/or | |
|recognise that expressions such as [pic] may represent 'partial |- Like Term/s that suits EVERY SECTION OF THE EXPRESSION | |
|factorisation' and that further factorisation is necessary to |[pic] | |
|'factorise fully' (Reasoning) [pic] |Combination of Common Integer and Pronumeral | |
| |Eg. | |
| |[pic] | |
| |Adjustment | |
| |Work with small integers and easy pronumerals | |
| |Recognising HCF and consistency of pronumerals within the expression given | |
| |Work as a many part factorising solution – don’t have to jump to the complete result | |
| |in first attempt. Make an easy obvious move first then TWEAK it further if can do!!! | |
|Expand binomial products and factorise monic quadratic expressions |Expanding |Factorising Monic Quadratics Questions |
|using a variety of strategies (ACMNA233) | |
|expand binomial products by finding the areas of rectangles, eg |Area Boxed Method Pre Thought From Stage 4 |s&topic_id=74&parent_name=Algebra%26nbsp%3B-%3E%2|
|[pic] | |6nbsp%3BFactoring+-%3E+Trinomials+with+a+%3D+1 |
|hence, |AREA Method |Expanding Double Brackets Jigsaw |
|[pic] |[pic] |
|use algebraic methods to expand binomial products, eg [pic] [pic] | |panding-double-bracket-jigsaw-6030126/ |
|factorise monic quadratic trinomial expressions, eg [pic] [pic] |Similar to a Magic Square Puzzle ~ Multiply the matching terms per ACTUAL box |Quadratic Sequences |
|connect binomial products with the commutative property of |position to get results. |
|arithmetic, such that [pic] (Communicating, Reasoning) [pic] |Write each of the results as your solution - before checking to see if any more |uadratic_sequences/revision/1/ |
|explain why a particular algebraic expansion or factorisation is |simplifying is necessary. |Huge Set of Questions Both Expanding and |
|incorrect, eg 'Why is the factorisation [pic] incorrect?' |Distrubution Rule Expansion CLAW Pre Thought From Stage 4 |Factorising |
|(Communicating, Reasoning) [pic] [pic] |Expanding Method for Binomials (2 brackets of 2 parts) |
| |[pic][pic] |/Factoring-Trinomials-to-Solve-Quadratic-Equation|
| |Take the FIRST position terms for both the brackets and multiply them |s-by-Rapalje.lesson |
| |Take the OUTER position terms from both brackets (making sure the +/- symbol is taken| |
| |as well) and multiply them |Factorising Trinomials |
| |Take the INNER position terms from both brackets (making sure the +/- symbol is taken|
| |as well) and multiply together |10.html |
| |Take the LAST position terms for both brackets (making sure the +/- symbol is taken | |
| |as well) and multiply them |
| |Once all multiplication of matched terms are done – check to see if any further |E5/LFacEq.htm |
| |simplifying needs to be done. | |
| | |
| |Factorising |uadratic_sequences/revision/1/ BBC Bitesize: |
| | |quadratic sequences – revision |
| |For each Quadratic to factorise the students need to consider that to find the |
| |factors to put into brackets. |ion.html TIMES Module 33: Number and Algebra: |
| |List at the factors of the LAST term |factorisation - teacher guide |
| |If its sign is POSITIVE both factors MUST have the SAME sign |
| |If its sign is NEGATIVE the factors MUST have DIFFERENT signs |gebraFour/ Algebra four |
| |Take the MIDDLE term and match it with the FACTORS and SIGNS | |
| |The sign of the MIDDLE term is the sign of the LARGEST factor | |
| |OR | |
| |Verbally discuss the idea of which two numbers MULTIPLY to get the LAST term | |
| |co-efficient | |
| |BUT also ADD or SUBTRACT to get the MIDDLE term co-efficient. | |
| |At first not even considering the direction of each value (-/+) | |
| |Once the values are decided set the Factor Brackets up as | |
| |[pic] | |
| |Factors could be 5 and 10 2 and 25 | |
| |Sign CHECK is a NEGATIVE therefore the factors have different signs | |
| |Therefore the numbers MUST be 2 and 25 | |
| |Negative 2 and Positive 25 | |
| | | |
| |[pic]then [pic] | |
| |Before finalising the factorisation with +/- [pic] | |
| | | |
| |Adjustment | |
| |Use factorising techniques to solve quadratic equations and draw graphs of parabolas | |
|Stage 5.3 - Algebraic Techniques § |5.2 Algebraic Fractions Pre Thought |Math In Living Color |
|Students: |[pic] |
|Add and subtract algebraic fractions with numerical denominators, | |20Living%20Color/math_in_living_c_o_l_o_r.htm |
|including those with binomial numerators | | |
|add and subtract algebraic fractions, including those with binomial| | |
|numerators, | | |
|eg [pic], [pic] | | |
|Expand binomial products using a variety of strategies (ACMNA233) |[pic] |Using FOIL Description |
|recognise and apply the special product, [pic] |[pic] | |
|recognise and name appropriate expressions as the 'difference of |[pic] |Boxed method expanding |
|two squares' (Communicating) [pic] [pic] |[pic] | |
|recognise and apply the special products, [pic] | | |
|recognise and name appropriate expressions as 'perfect squares' | |Variety of Special Products |
|(Communicating) [pic] [pic] | | |
|use algebraic methods to expand a variety of binomial products, | | |
|including the special products, eg [pic], [pic] | | |
|simplify a variety of expressions involving binomial products, | | |
|eg [pic], [pic] | | |
|Factorise monic and non-monic quadratic expressions (ACMNA269) |Methods of Factorising |Videos from Khan Academy Quadratic Factorising |
|factorise algebraic expressions, including those involving: [pic] |Common factors (see above in 5.2) |ALL GOOD!!! |
|common factors | |
|a difference of two squares |Difference of Two Squares |s |
|grouping in pairs for four-term expressions |[pic] [pic] |(You could even get the students to check these |
|perfect squares |Grouping in Pairs |out) |
|quadratic trinomials (monic and non-monic) |[pic] | |
|use a variety of strategies to factorise algebraic expressions, |therefore [pic] | |
|eg [pic], [pic], [pic], [pic], [pic] [pic] |Perfect Squares |Factorising Questions |
|factorise and simplify complex algebraic expressions involving |[pic] |
|algebraic fractions, |Trinomials | |
|eg [pic], [pic], [pic], [pic] |[pic] |Factorising Many Quadratic Questions |
| |[pic] |
| |[pic] | |
| |[pic] | |
| |[pic] | |
| |[pic] |Math In Living Color |
| |[pic] |
| |[pic] |20Living%20Color/math_in_living_c_o_l_o_r.htm |
| |[pic] [pic] |Complex Fraction PowerPoint |
| |[pic] |
| | |actions |
| |[pic] |Further Complex Lesson Connected Here |
| | |
| | |3dy5nbGVuY29lLmNvbS9zZWMvbWF0aC9hbGdlYnJhL2FsZ2Vi|
| | |cmExL2FsZ2VicmExXzAzL3N0dWR5X2d1aWRlL3BkZnMvYWxnM|
| | |V9wc3NnX0cwOTgucGRmCkxlc3NvbiA4IC0gTWl4ZWQgRXhwcm|
| | |Vzc2lvbnMgYW5kIENvbXBsZXggRnJhY3Rpb25zIC0gR2xlbmN|
| | |vZQ== |
|Registration |Evaluation |
|Class: __________________________ |Teachers evaluate the extent to which the planning of the unit has remained focused on the syllabus outcomes. After the unit has been |
|Start Date: _______________________ |implemented, there should be opportunity for both teachers and students to reflect on and evaluate the degree to which students have |
|Finish Date: ______________________ |progressed as a result of their experiences, and what should be done next to assist them in their learning. |
|Teacher’s Signature: _______________________ |Evaluation reflects: |
| |The effectiveness of the program in meeting the diverse needs of students and identifies curriculum adjustments |
| |Level to which syllabus outcomes have been demonstrated by students |
| |The effectiveness of pedagogical practices employed |
| |Suggested program adjustments |
| |Elements of the school’s Contemporary Learning Framework |
Sample questions
Highlight the response that best describes your view to the following statements and provide comments in the spaces provided.
1. The set text/s (if relevant) were suitable for the student needs and interests:
|STRONGLY AGREE |AGREE |UNSURE |STRONGLY DISAGREE |
2. There were sufficient and suitable resources to teach the unit:
|STRONGLY AGREE |AGREE |UNSURE |STRONGLY DISAGREE |
3. There was sufficient time to teach the set content:
|STRONGLY AGREE |AGREE |UNSURE |STRONGLY DISAGREE |
4. Assess the degree to which syllabus outcomes have been demonstrated by students in this unit:
.......................................................................................................................................................................................................................
.......................................................................................................................................................................................................................
.......................................................................................................................................................................................................................
.......................................................................................................................................................................................................................
5. Evaluate the degree to which the diverse needs of learners have been addressed in this unit:
.......................................................................................................................................................................................................................
.......................................................................................................................................................................................................................
.......................................................................................................................................................................................................................
.......................................................................................................................................................................................................................
6. Comment on the effectiveness of pedagogical practices employed in this unit:
......................................................................................................................................................................................................................
......................................................................................................................................................................................................................
......................................................................................................................................................................................................................
......................................................................................................................................................................................................................
7. Assessment was meaningful and appropriate to reflect student learning and achievement:
.......................................................................................................................................................................................................................
.......................................................................................................................................................................................................................
.......................................................................................................................................................................................................................
.......................................................................................................................................................................................................................
8. Suggested program adjustments / other comments:
.......................................................................................................................................................................................................................
.......................................................................................................................................................................................................................
.......................................................................................................................................................................................................................
.......................................................................................................................................................................................................................[pic]
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.