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|MA5.2-6NA Algebraic Techniques | Mathematics Stage 5.1/5.2 Year 9 2014 |
|Summary of Sub Strands |Duration 3.5 weeks |
|S4 Algebraic Techniques 1 & 2 |Start Date: |
| |Completion Date |
| |Teacher and Class: |
|Unit overview |Outcomes |Big Ideas/Guiding Questions |
|Uses the algebraic symbol system to simplify, |MA5.2-6NA simplifies algebraic fractions, and | |
|expand and factorise simple algebraic |expands and factorises quadratic expressions | |
|expressions | | |
|Simplifies, expands and factorises algebraic | | |
|expressions involving fractions and negative | | |
|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 notation |
| | |evaluate, factorise, fractional / negative / zero index, highest common factor, like terms, lowest |
| | |common denominator / multiple, power, pronumeral, reciprocal, simplify, square / cube root, |
| | |substitute |
| | |Expand, Binomial, Quadratic, simultaneous, factorising, etc. |
|Catholic Perspectives |School Free Design |
|MacKillop College Bathurst is a Catholic faith community, dedicated to the education of young women. |This is a free design area for schools to add local additional areas. This could include: |
|The Mathematics teachers undertake to uphold the ethos and teachings of the Catholic church and to |Context if you prefer the unit overview and context to be separate |
|support the liturgical life of the College. They promote in the classroom a sense of compassion, |School focus for learning – eg blooms taxonomy, solo taxonomy, contemporary learning, habits of mind,|
|respect between students and staff and a positive and supportive learning environment. |BLP (building learning power) |
|Numeracy operates within a variety of social contexts. From a Catholic perspective, numeracy must be |Any specific social and emotional learning which could be embedded into the unit eg enhanced group |
|infused with a vision of the innate dignity of all students, as created in the image and likeness of |work |
|a loving, generous and creating God. Teachers of Mathematics 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. | |
|Assessment Overview |
|Quizzes |
|Online tests |
|Assignment |
|Content |Teaching, learning and assessment |Resources |
|At the end of this unit the student should be able to: |Fractions Pre Thoughts from Stage 4 |Math In Living Color ~ Instructions and Questions|
|Stage 5.2 - Algebraic Techniques | |
|Students: |Simplifying Expressions Pre Thoughts From Stage 4 |20Living%20Color/math_in_living_c_o_l_o_r.htm |
|Apply the four operations to simple algebraic fractions with | | |
|numerical denominators (ACMNA232) |HCF for Integers and Pronumerals Pre Thoughts From Stage 4 | |
|simplify expressions that involve algebraic fractions with | | |
|numerical denominators, |Addition and Subtraction | |
|eg [pic] [pic] [pic] [pic] |[pic] | |
|connect the processes for simplifying expressions involving |[pic] | |
|algebraic fractions with the corresponding processes involving | | |
|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 | |
| |Take the INNER position terms from both brackets (making sure the +/- symbol is taken| |
| |as well) and multiply together | |
| |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 | |
| |simplifying needs to be done. | |
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| |Factorising | |
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| |For each Quadratic to factorise the students need to consider that to find the | |
| |factors to put into brackets. | |
| |List at the factors of the LAST term | |
| |If its sign is POSITIVE both factors MUST have the SAME sign | |
| |If its sign is NEGATIVE the factors MUST have DIFFERENT signs | |
| |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. |Factorising Trinomials |
| |At first not even considering the direction of each value (-/+) |
| |Once the values are decided set the Factor Brackets up as |10.html |
| |[pic] | |
| |Factors could be 5 and 10 2 and 25 |
| |Sign CHECK is a NEGATIVE therefore the factors have different signs |E5/LFacEq.htm |
| |Therefore the numbers MUST be 2 and 25 | |
| |Negative 2 and Positive 25 |
| | |uadratic_sequences/revision/1/ BBC Bitesize: |
| |[pic]then [pic] |quadratic sequences – revision |
| |Before finalising the factorisation with +/- [pic] |
| | |ion.html TIMES Module 33: Number and Algebra: |
| |Adjustment |factorisation - teacher guide |
| |Use factorising techniques to solve quadratic equations and draw graphs of parabolas |
| | |gebraFour/ Algebra four |
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