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|Stage 5.3 - Further Linear equations |
|Summary of Substrands |Duration: 4 weeks |
|S4 Equations |Start Date: |
|Non Linear Relationships and Graphing | |
|Binomial Products | |
| |Completion Date: |
| |Teacher and Class: |
|Outcomes |
|MA5.2-1WM selects appropriate notations and conventions to communicate mathematical ideas and solutions |
|MA5.2-2WM interprets mathematical or real-life situations, systematically applying appropriate strategies to solve problems |
|MA5.2-3WM constructs arguments to prove and justify results |
|MA5.2-8NA solves linear and simple quadratic equations, linear inequalities and linear simultaneous equations, using analytical and graphical techniques |
|MA5.3-1WM uses and interprets formal definitions and generalisations when explaining solutions and/or conjectures |
|MA5.3-2WM generalises mathematical ideas and techniques to analyse and solve problems efficiently |
|MA5.3-3WM uses deductive reasoning in presenting arguments and formal proofs |
|MA5.3-7NA solves complex linear, quadratic, simple cubic and simultaneous equations, and rearranges literal equations |
|Overview |Key Words |Suggested Assessment |
|Students will revisit solution of linear equations and investigate |Linear equation, grouping symbols, denominator, coefficient, | |
|the solution of inequations and what this means in terms of plotting|quadratic, solution, solve, pronumerals, inverse, variable, | |
|a solution. They will solve quadratic equations of the form ax2 + bx|unknown, substitute, quadratic equation, expression, operation, | |
|+ c = 0, with both rational and irrational solutions. They will |elimination, substitution, cubic, parabola, subject, | |
|solve linear and nonlinear simultaneous equations using graphical |restrictions, inequation, inequality, number plane, number line, | |
|methods, substitution and elimination methods. They will investigate|rational, irrational, exact form, approximation | |
|solutions to cubic equations & changing the subject of a formula is | | |
|introduced. | | |
|Rego |Content |Teaching, learning and assessment |Resources |
| |Students: |General Note: Students must be encouraged as a matter of routine to always check their | |
| |Solve linear equations (ACMNA215) |solutions by substitution | |
| |solve linear equations, including equations that involve| | |
| |grouping symbols, |The first two dot points are revision and extension of Stage 4 content, so review processes |Resources as outlined in Stage 4 Simple Equations |
| |eg [pic] [pic] |of backtracking, balancing etc. Before extending into harder examples. |may be used as a revision tool + from the |
| |Solve linear equations involving simple algebraic |The concept of lowest common multiple will need to be revisited to solve algebraic equations|following websites |
| |fractions (ACMNA240) |involving fractions. Also discuss if you cannot find the LCM just multiple the denominators |
| |solve linear equations involving one or more simple |of the fraction, stressing that you will need to look at simplifying your answer. |swf |
| |algebraic fractions, |Also it is important to stress that if there is more than one term in the numerator, |balancing equations |
| |eg [pic] [pic] [pic] [pic] [pic] |grouping symbol need to be inserted to ensure the correct sign of coefficient is obtained | |
| | |when multiplying by the LCM. |simple graph creation |
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| | |Adjustments |Free online graphing calculator |
| | |Lower ability students will need fractional equations to have smaller denominators, as their|
| | |table skills may be poor and they may experience difficulty finding the LCM of larger |ndex.htm |
| |compare and contrast different algebraic techniques for |numbers. |Khan Academy (videos) - |
| |solving linear equations and justify a choice for a |To extend students graph y equal to the equation and solve the equation graphically. | (can sign in |
| |particular case (Communicating, Reasoning) [pic] |Graphing software Geogebra, sketchpad and quickgraph can assist students to graph. |using google) |
| | | |Manga High |
| |Solve complex linear equations involving algebraic | | |
| |fractions | | |
| |solve a range of linear equations, including equations | | |
| |that involve two or more fractions, | | |
| |eg [pic], [pic] | |Interactive solving equations |
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| |Substitute values into formulas to determine an unknown | | |
| |(ACMNA234) |Revise LCM and then introduce the concept of multiplying every term in the equation by the | - in a game form |
| |solve equations arising from substitution into formulas,|LCM to eliminate the denominator. | |
| |eg given [pic] and [pic], [pic], solve for [pic] |Thus employ methods of solving linear equations. | |
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| |substitute into formulas from other strands of the | | |
| |syllabus or from other subjects to solve problems and | | |
| |interpret solutions, eg [pic] [pic] [pic] [pic] (Problem| | |
| |Solving) [pic] | | |
| | | | |
| |Solve problems involving linear equations, including | | |
| |those derived from formulas (ACMNA235) | | |
| |translate word problems into equations, solve the | |Substitution quizzes |
| |equations and interpret the solutions | |
| |state clearly the meaning of introduced pronumerals when| |k8/bk8i12/bk8_12i1.htm |
| |using equations to solve word problems, | | |
| |eg 'n = number of years' (Communicating) [pic] |Use formulae that the students have seen before so that they can see the relevance of their | |
| |solve word problems involving familiar formulas, eg 'If |solutions, e.g. perimeter and area of shapes. Student must be vigilant that units are the |
| |the area of a triangle is 30 square centimetres and the |same when substituting into a formula. Stress that the answers may have relevant units e.g. |/index.php |
| |base length is 12 centimetres, find the perpendicular |cm2 |M009037 |
| |height of the triangle' (Problem Solving) [pic] | | |
| |explain why the solution to a linear equation generated | | |
| |from a word problem may not be a solution to the given | |S3262 – using spreadsheets with equations article |
| |problem (Communicating, Reasoning) [pic] | | |
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| |Solve linear inequalities and graph their solutions on a| | |
| |number line (ACMNA236) | | |
| |represent simple inequalities on the number line, | | |
| |eg represent [pic] on a number line | | |
| |recognise that an inequality has an infinite number of | | |
| |solutions unless other restrictions are made | | |
| |solve linear inequalities, including through reversing | |
| |the direction of the inequality sign when multiplying or| |ntable/53491.html |
| |dividing by a negative number, and graph the solutions, | |bank of word problems |
| |eg solve and graph the inequalities on a number line of | | |
| |[pic] | | |
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| | |Initially students will need extensive practice converting between words and the | |
| |use a numerical example to justify the need to reverse |corresponding algebraic equations. It is recommended that “sensible” pronumerals be used | |
| |the direction of the inequality sign when multiplying or|e.g. n = number etc. | |
| |dividing by a negative number (Reasoning) [pic] | |M012159 – links solving linear equations to a |
| | |Adjustment: Less able students may only graduate to simple 2 step equations. |graph |
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| | |Advisable to review the skills involved in rearranging an equation to have the unknown on | |
| |verify the direction of the inequality sign by |the LHS | |
| |substituting a value within the solution range |Student must be vigilant that units are the same when substituting into a formula. Stress | |
| |(Reasoning) [pic] |that the answers may have relevant units e.g. cm2 | |
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| |Solve linear simultaneous equations, using algebraic and| | |
| |graphical techniques, including with the use of digital | | |
| |technologies (ACMNA237) | | |
| |solve linear simultaneous equations by finding the point| | |
| |of intersection of their graphs, with and without the | | |
| |use of digital technologies [pic] |NOTE : Real world solutions generally involve positive answers. |Inequalities |
| | | |M010538 |
| |solve linear simultaneous equations using appropriate | | |
| |algebraic techniques, including with the use of the | | |
| |'substitution' and 'elimination' methods, eg solve [pic]| |Powerpoints, lesson plans, worksheets available on|
| | | |TES Australia |
| |select an appropriate technique to solve particular |Start with x > 5 and get the students to give an answer. Then discuss that the inequality | |
| |linear simultaneous equations by observing the features |sign produces an infinite number of solutions. Graph the solutions on a number line, | |
| |of the equations (Problem Solving) [pic] [pic] |emphasizing | |
| | | | |
| |generate and solve linear simultaneous equations from |Links to > or < sign | |
| |word problems and interpret the results [pic] [pic] | |
| | |Links to ≤ or ≥ sign |ities/ |
| | |Some examples where restrictions apply need to be supplied and investigated | |
| | | | |
| |Solve simple quadratic equations using a range of | | |
| |strategies (ACMNA241) |Start by comparing and contrasting simple equations and inequations that do not require | |
| |solve simple quadratic equations of the form [pic], |reversal of inequality sign. | |
| |leaving answers in exact form and as decimal |x + 1 = 9 has one solution | |
| |approximations |x + 1 > 9 has an infinite number of solutions( need to show this on a number line. | |
| |explain why quadratic equations could be expected to | | |
| |have two solutions (Communicating, Reasoning) [pic] |Adjustment : Higher ability students may graph the inequality on a number plane to | |
| |recognise and explain why [pic] does not have a solution|illustrate the result | |
| |if [pic] is a negative number (Communicating) [pic] | | |
| |solve quadratic equations of the form [pic], limited to | | |
| |[pic], using factors | | |
| | | | |
| |connect algebra with arithmetic to explain that if | | |
| |[pic], then either [pic] or [pic] (Communicating, | | |
| |Reasoning) [pic] |After comprehensive consolidation of the above, use simple numerical examples together with | |
| |check the solution(s) of quadratic equations by |reiteration of equation solving strategies (same thing to both sides) to illustrate the need| |
| |substitution (Reasoning) [pic] |to change the inequality sign. | |
| | |e.g. | |
| |Solve a wide range of quadratic equations derived from a|5 > 2 true ( multiply both side by 4 | |
| |variety of contexts (ACMNA269) |20> 8 true ( subtract 5 | |
| |solve equations of the form [pic] by factorisation and |15> 3 true ( divide by -3 | |
| |by 'completing the square' |-5> -1 NO!!!!! | |
| | |This leads to recognition of reversing the sign when multiplying or dividing by a negative | |
| | |number | |
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| |use the quadratic formula [pic] to solve quadratic |Students must be encouraged as a matter of routine to always check their solutions by |ml- algebraic explanation |
| |equations |substitution |
| |solve a variety of quadratic equations, eg [pic], | |ons.htm - algebraic and graphic |
| |[pic], [pic], [pic] | |M010538 |
| |choose the most appropriate method to solve a particular| | |
| |quadratic equation (Problem Solving) [pic] | | |
| |check the solutions of quadratic equations by | | |
| |substituting [pic] |Adjustment : Recap sketching linear equations if needed. | |
| |identify whether a given quadratic equation has real |Use two rulers as the linear equations and get the students to arrive at the possibilities | |
| |solutions, and if there are real solutions, whether they|of how these two lines can intersect. Then discuss the terminology of two lines intersecting|
| |are or are not equal [pic] |and solving the equations simultaneously. |s-Equations-Magic-Trick-6302253/ |
| |predict the number of distinct real solutions for a |Solving linear equations graphically with supplied graphs, and students practicing reading | |
| |particular quadratic equation (Communicating, Reasoning)|off the point of intersection, i.e. the solution to solving the equations simultaneous. |Elimination method |
| |[pic] |Note that the solution to a pair of simultaneous equations will give you two values and that|
| | |it is a coordinate on the number plane. |imination-method/nsw/elimination-method/algebra/si|
| |connect the value of [pic] to the number of distinct |Once proficient with reading solutions off a graph, students then move on to graphing their |multaneous-equations |
| |solutions of [pic] and explain the significance of this |own straight lines. Note lines must be graphed accurately and not sketched. | |
| |connection (Communicating, Reasoning) [pic] | | |
| |solve quadratic equations resulting from substitution |Substitution method first. Begin with example like | |
| |into formulas |a = 4b + 2 | |
| |create quadratic equations to solve a variety of |a= 3b - 4 | |
| |problems and check solutions [pic] |Multiple examples of this type will be needed for students to become proficient | |
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| | |Illustrate through examples that the substitution method is not always the best choice. | |
| | |Link between substitution and elimination methods could be a common pair of simultaneous | |
| | |equations | |
| | |Demonstrate the elimination method with students completing multiple examples until | |
| | |proficient | |
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| |explain why one of the solutions to a quadratic equation|Selection of the best method to use needs to be addressed as each method is taught and | |
| |generated from a word problem may not be a possible |learned. Particular features that would lead to the selection of a particular method need to| |
| |solution to the problem (Communicating, Reasoning) [pic]|be noted. | |
| | |This could be shown through the completion of example using both methods | |
| | | |Generates quadratic equations and checks online |
| |substitute a pronumeral to simplify higher-order |Generate simple example of simultaneous equation word problems. |
| |equations so that they can be seen to belong to general | |-equations-10qqi.html |
| |categories and then solve the equations, eg substitute |Adjustment: Higher ability level students could graduate to linear programming type examples| |
| |[pic] for [pic] to solve [pic] for [pic] |e.g. cost, profit, break-even point |Null Factor Law |
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| |Solve simple cubic equations |Assessment | |
| |determine that for any value of [pic] there is a unique |Pretest on square numbers and square roots of perfect squares. | |
| |value of [pic] that solves a simple cubic equation of | | |
| |the form [pic] where [pic] |The pretest can then lead into looking at the square root of a number that is not a perfect | |
| |explain why cubic equations of the form [pic] where |square and review of Stage 4 content | |
| |[pic] have a unique solution (Communicating, Reasoning) |x 2= a. At this point integer solutions to x 2= a must be considered as in Stage 4 only | |
| |[pic] |positive solutions were considered. | |
| |solve simple cubic equations of the form [pic], leaving | | |
| |answers in exact form and as decimal approximations |Adjustment |Online quiz quadratics |
| | |For more able students link x 2= a to the graph of a parabola to emphasis both the negative|
| | |and positive solution. This will allow the students to see that x 2= a does not have a |n/factoring.quiz |
| | |solution if a < 0. |Grouping in pairs |
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| | |Review negative times negative = positive, therefore will never get the equation x2 = a |
| | |negative number |w-to-factor-by-grouping.php |
| |Rearrange literal equations |It is key for students to recognize that the graph of an equation of this form is parabolic.| |
| |change the subject of formulas, including examples from |This can be achieved through investigation of various graphs using graphing software |Completing the square |
| |other strands and other learning areas, |E.g. Graph and discuss y= x2 – x – 6 and y= (x-3)(x+2) and “discovering” they are the same |
| |eg make a the subject of [pic], make r the subject of |graph. Note that the second is a factorised version of the first and is helpful when trying |ksheets/Quadratic%20Equations%20By%20Completing%20|
| |[pic], make b the subject of [pic] |to both sketch and solve. |the%20Square.pdf |
| |determine restrictions on the values of variables |Link that the solution of (x-3)(x+2) = 0 actually gives us the x-intercepts. Discuss the |
| |implicit in the original formula and after rearrangement|term “Null Factor Law” |the-square-math.php |
| |of the formula, eg consider what restrictions there |Reinforce this idea by initially solving and sketching already factorsied forms of |
| |would be on the variables in the equation [pic] and what|quadratics |/completing_the_square/v/completing-the-square-to-|
| |additional restrictions are assumed if the equation is |Using graphing software get the students to graph a number of quadratic equations of the |solve-quadratic-equations |
| |rearranged to [pic] (Communicating, Reasoning) [pic] |form y = ax2 + bx + c, discuss the shape of the graph and what it means to solve ax2 + bx +| |
| | |c = 0, i.e. that we are finding the x-intercepts. |Quadratic formula |
| |Solve simultaneous equations, where one equation is |Adjustments |
| |non-linear, using algebraic and graphical techniques, |Less able students may only be capable of solving quadratic equations algebraically when |ksheets/Quadratic%20Formula.pdf |
| |including the use of digital technologies |given in factorized form. | |
| |use analytical methods to solve a variety of |Use PSN or grouping in pairs to solve ax2 + bx + c = 0, explaining that if p × q = 0, then | |
| |simultaneous equations, where one equation is |either p = 0 or q = 0. | |
| |non-linear, | | |
| |eg [pic], [pic], [pic] | | |
| |choose an appropriate method to solve a pair of | | |
| |simultaneous equations (Problem Solving, Reasoning) | | |
| |solve pairs of simultaneous equations, where one | | |
| |equation is non-linear, by finding the point of |As the quadratic can be expressed in a variety of ways and the coefficient of x2 is not | |
| |intersection of their graphs using digital technologies |limited to a= 1, we must consider other methods of solving quadratics including | |
| |determine and explain that some pairs of simultaneous |psn | |
| |equations, where one equation is non-linear, may have no|grouping in pairs | |
| |real solutions (Communicating, Reasoning) [pic] |completing the square | |
| | |quadratic formula | |
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| | |These methods need to be practiced extensively and discussion as to when to use with method | |
| | |is essential | |
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| | |Investigate the quadratic formula and indicate what constraints exist | |
| | |a negative results under the square root sign in the quadratic formula, i.e. no solution. | |
| | |a perfect square appears under the square root sign in the quadratic formula, i.e. one | |
| | |solution. | |
| | |A positive integer that is not a perfect square appears under the square root sign in the | |
| | |quadratic formula, i.e. two solutions | |
| | |This can be an informal discussion which will be tied together below . | |
| | |This discussion will introduce new terminology: real, unreal, discriminant, equal roots, | |
| | |rational, irrational, definite, indefinite. | |
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| | |Adjustment: Higher ability students can link real and unreal solutions to a graphical | |
| | |interpretation. | |
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| | |Note : Success with this section will mean recapping of sketching parabolas and solution of | |
| | |quadratics and their links to each other | |
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| | |Guide students through careful selection of examples to discover the concept of the | |
| | |discriminant. It is not appropriate to use this term as a matter of course but introduction | |
| | |can lead to some very rich mathematical discussion | |
| | |Adjustment: Link once again to graph of parabola and where it sits in relation to x axis | |
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| | |Students to complete increasingly more difficult examples e.g. | |
| | |When a number is added to its square the result is 90. Find. the number. | |
| | |Twice a number is added to three times its square. If the result is 16, find the number. |Changing the subject of an equation |
| | |Five times a number is added to two times its square. If the result is 168, find the number.| |
| | |A practical problem involving perimeter and area of a rectangle can be investigated to show |
| | |a real life situation. |mages/Microsoft_Word_-_Changing_the_Subject_of_an_|
| | |The school wants to build a rectangular vegetable garden with dimensions x metres by y |equation.pdf |
| | |metres with a perimeter of 12 metres. Find the dimensions that will maximize the area of the| |
| | |vegetable garden. | |
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| | |Adjustment: The level of difficulty and format of questions involving substitution of a | |
| | |pronumeral to simply higher-order equations will depend on the ability of students. | |
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| | |Begin with factorised equations (x2)2 – 13(x2) + 36 = 0 and move to unfactorised equations | |
| | |x4 – 13x2 + 36 = 0. | |
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| | |Adjustment: Some students may only be able to solve these type of equations with teacher | |
| | |assistance. | |
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| | |This section should not be attempted until students display a proficiency with quadratics. | |
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| | |Discuss the opposite of squaring is square rooting, so what is the opposite of cubing a | |
| | |number? | |
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| | |Start with solving simple cubics x3 = k that will produce integer solutions. | |
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| | |Adjustment: Discuss why the cube root of a negative number can be found but that a negative | |
| | |number cannot be square rooted. This can lead to looking at positive and negative numbers | |
| | |and if even powers and/or odd powers can produce these numbers. | |
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| | |Once students have grasped solving simple cubics move to introducing a coefficient to x3 | |
| | |i.e. equations of the form ax3 = k.. | |
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| | |Adjustments: The more able students can use software to graph simple cubics and discus how | |
| | |their answers can be obtained graphically. | |
| | |Ensure that questions are graded when changing the subject of an equation so that all | |
| | |students experience a level of success. Stress that changing the subject of the equation | |
| | |requires reversing steps, so inverse operations are required. | |
| | |Recapping inverse operations for squaring, cubing, square roots etc. should be done prior to| |
| | |attempting harder questions. | |
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| | |Using formulas from other strands will reinforce formula knowledge. | |
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| | |The importance of checking solutions for restrictions is essential. Is zero or a negative | |
| | |number applicable to the situation? | |
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| | | |Solving simultaneous equations -linear and |
| | |This should be introduced graphically initially, stressing that the point(s) of intersection|non-linear |
| | |are the solution(s) to solving simultaneous equations. | |
| | |Look at a variety of cases of non-linear and linear simultaneous equations and discuss how | |
| | |many solutions are possible. This can be done on the whiteboard, by drawing a parabola, | |
| | |hyperbola, cubic etc. and getting students to place the metre ruler across the diagram to | |
| | |show the number of possible solutions. | |
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| | |Adjustment: Lower ability students may not progress to algebraic techniques. |Free graphing software |
| | | |Geogebra |
| | |Ensure that examples are scaffolded and that guidance is provided to select the most |Quickgraph |
| | |efficient method to solve the equations simultaneously. As the students’ confidence in | |
| | |algebraic manipulation increases, move towards more challenging questions. |Additional Resources |
| | | |Quadratics and cubics |
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| | | |uadratic-equations |
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| | | |asp?section=algebra§ionID=2&topic=quadratics |
| | |Stress: | |
| | |one solution – one point of intersection | |
| | |two solution – two points of intersection | |
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