POST-PRIMARY Mathematical Studies ad Applicatios: Mathematics, Busiess Studies Guidelies for Teachers of Studets with MILD Geeral Learig Disabilities
Cotets Itroductio 3 Approaches ad methodologies 4 Exemplars 20 Appedix 151
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY Itroductio These guidelies are desiged to support teachers of studets with mild geeral learig disabilities who are accessig the juior cycle curriculum i the area of Mathematics. They are part of a suite of guidelies produced by the Natioal Coucil for Curriculum ad Assessmet with a focus o special educatioal eeds. Each set of guidelies correspods to a area of experiece of the Juior Cycle curriculum ad offers exemplars of good classroom practice i support of the kowledge ad skills associated with that area of experiece. These guidelies are is desiged to support the teacher of Mathematical Studies ad Applicatios for studets with special educatioal eeds, withi the cotext of a whole school pla. I additio to the guidelies preseted here, similar materials have bee prepared for teachers workig with studets accessig the Primary School Curriculum. Cotiuity ad progressio are importat features of the educatioal experiece of all studets, but they are particularly importat for studets with special educatioal eeds. Therefore, all the exemplars preseted here iclude a referece to opportuities for prior learig i the Primary School Curriculum. The exemplars have bee prepared to show how studets with mild geeral learig disabilities ca access the curriculum through differetiated approaches ad methodologies. It is hoped that these exemplars will eable teachers to provide further access to the other areas of the curriculum. A rage of assessmet strategies is idetified i order to esure that studets ca receive meaigful feedback ad experiece success i learig.
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY Approaches ad methodologies I Approaches ad Methodologies idividual differeces are emphasised, ad potetial areas of difficulty ad implicatios for learig are outlied ad liked with suggestios for approaches ad methodologies for classroom use. Idividual differeces i talets, stregths, ad eeds All studets will beefit from a variety of teachig styles ad classroom activities. Studets with mild geeral learig disabilities will beefit particularly if the teacher is aware of their idividual talets, stregths, ad eeds before embarkig o a ew activity. Cosultatio ad/or ivolvemet i the Idividual Educatio Plaig process as well as teacher observatio will assist the teacher of Mathematical Studies ad Applicatios i orgaisig a appropriate learig programme for a studet with mild geeral learig disabilities. Such a approach will assist the teacher i selectig suitably differetiated methods for the class. If learig activities are to be made meaigful, relevat, ad achievable for all studets the it is the role of the teacher to fid ways to respod to that diversity by usig differetiated approaches ad methodologies. This ca be achieved by esurig that objectives are realistic for the studets settig short ad varied tasks esurig that the learig task is compatible with prior learig providig opportuities for iteractig ad workig with other studets i small groups allowig studets to sped more time o tasks orgaisig the learig task ito small stages esurig that the laguage used is pitched at the studets level of comprehesio ad does ot hider uderstadig the activity usig task aalysis i outliig the steps to be leared/completed i ay give task modellig task aalysis by talkig through the steps of a task as it is beig doe posig key questios to guide studets through the stages/processes, ad to assist i self-directio ad correctio
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY usig graphic symbols as remiders to assist studets uderstadig of the sequece/steps i ay give task/problem creatig a cogeial learig eviromet by usig cocrete ad, where possible, everyday materials, ad by displayig word lists, lamiated charts with pictures. It is importat to teach mathematical laguage actively to the studets with mild geeral learig disabilities ad to reiforce it o a daily basis, sice they may have geeral laguage difficulties. A mathematical dictioary of words, symbols, ad diagrams ca be kept by the teacher ad by the studets themselves ad ca be preseted as a wall chart display. Mathematical Laguage Mathematics should be see as a laguage with its ow vocabulary of both words ad symbols. May studets cofuse mathematical laguage with ordiary laguage. They say He s bigger tha me whe they mea older, or My table is loger tha his whe they mea wider. It is importat to teach this laguage actively to the studets ad to reiforce it o a daily basis. Studets will eed to be exposed to mathematical laguage ad have it reiforced at a receptive level i a variety of situatios before they will develop the ability to use it themselves. The vocabulary of mathematics, symbols, ad tools are used i particular circumstaces. I geeral the studet is ulikely to hear or read much mathematical laguage outside the classroom. The teacher, as the mediator betwee the studet ad the world of mathematics, eeds to examie the classroom use of mathematical laguage carefully. Is it cosistet, accurate, ad uambiguous? Will studets experiece the same use of laguage as they move from oe class to aother? Are studets able to use appropriate mathematical laguage precisely? Ca studets relate some mathematical laguage to real-world situatios? Such errors ca be avoided if a rage of differet examples is used to explai a cocept. For example, if a studet is taught that Figure 1 is a right-agled triagle ad promptly labels Figure 2 a left-agled triagle the presetatio of a rage of triagles i differet orietatios ca help correct the miscoceptio. Figure 1 Figure 2 Teachers eed to be aware of the dager of usig mathematical tricks ad short cuts. Certai phrases have become commo i the mathematics classroom, particularly i the areas of arithmetic ad algebra, but they ofte serve to coceal the cocept behid what is occurrig. Laguage that describes trasformatios i terms of the surface structure oly should be avoided, because it focuses attetio o the form rather tha the meaig which gives rise to the trasformatio. Examples of this would be: Take it over to the other side ad chage the sig. Cross multiply. Move the decimal poit over. Tur it upside dow ad multiply. Collect all the x s o oe side of the equatio. Always do to the top what you do to the bottom. To multiply by te add a ought. These descriptios tell someoe oly what to do, with the result that there is little impetus to examie them to see why they might be helpful. Cosistecy of approach is vital. It is importat that all teachig staff who are i cotact with the studet ad parets are aware of the termiology beig used. For example, where appropriate plaig has occurred the mathematics teacher ca use the laguage of measuremet (legth, perimeter, area, loger, shorter, etc.) while at the same time the Eglish teacher ca adopt reiforcemet activities that icorporate the same words ad the sciece ad/or woodwork teacher ca egage the studets i practical measurig activities. Parets ca also be ecouraged to use the words at home. Keepig parets iformed of the words beig used ad the importace of usig them frequetly will help the studet to use them i real cotexts.
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY The followig is a example of a ote that could be set to parets: This week s keywords i mathematics are: measure, size, legth, distace, uits, cetimetre, metre, kilometre, loger, shorter, how log? Please use them ofte ad ecourage your child to use them i appropriate situatios. Studets may have difficulties with mathematical symbols. Charts i the classroom showig the symbol, the word, a example, ad a diagram ca help to reiforce a correct iterpretatio of a symbol. Where possible a real-world lik to the symbol should also be icluded. Studets ca build up a dictioary of symbols as they progress through the course. Symbol Word Example Diagram Real-world itersectio A B={2,3} parallel X Y A road itersectio is where two roads cross each other. Trai tracks go o without ever meetig. Backgroud to the Juior Certificate Foudatio Level mathematics course These guidelies are writte i the cotext of the area of experiece Mathematical Studies ad Applicatios as outlied i the Juior Cycle Review (1999). The topics referred to i the exemplar material are draw from the foudatio level of the Juior Certificate mathematics course. Liks to the relevat primary school topics ad the correspodig statemets i the Juior Certificate School Programme are metioed. This sectio outlies the ratioale, aims, ad cotet of the foudatio level mathematics course. The foudatio level Juior Certificate mathematics course is desiged for studets who are ot ready for, or who are usuited to the ordiary level course. Studets may ot be ready to deal with some of the more abstract mathematical cocepts; they may be fidig the trasitio from primary to post-primary school particularly difficult; or they may have learig styles that are ot met by the traditioal approach at post-primary level. This challeges teachers to exted ad diversify their teachig styles. These studets still eed to lear mathematics to help them i everyday life (social mathematics), further study or traiig (vocatioal mathematics), or perhaps to ehace their thikig skills ad problem solvig skills. The foudatio level course is desiged to help the studet to costruct a clearer kowledge of basic mathematics develop improved skills i basic mathematics develop a awareess of the usefuless of mathematics feel she/he is makig progress through the itroductio of ew material egage i a rage of learig styles through, for example, the visual, spatial, ad umerical aspects of mathematics.
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY The particular target group may respod well to activities that improve studets self-cofidece (I ca do mathematics) The assessmet objectives, also stated i the syllabus, deal with mathematical kowledge, uderstadig, ad applicatio, ad with the studet s psychomotor skills ad the ability to commuicate what they are learig. improve studets cofidece i the subject (mathematics makes sese, mathematics is useful) support the acquisitio ad cosolidatio of fudametal skills embed mathematics i meaigful cotexts create opportuities for studets to experiece success create opportuities for studets to reflect o their ow experiece ad performace. The cotet of the Primary School Curriculum is take as the prerequisite for studets followig the foudatio level course. As will be see from the exemplars that follow, may of the primary strads have bee revised ad treated i greater depth. I this way there is a atural cotiuity from the primary curriculum to the post-primary curriculum. The specific aims of the foudatio level course, as stated i the syllabus, are that the course will provide studets with a uderstadig of the basic mathematical cocepts ad relatioships cofidece ad competece i basic skills the ability to solve simple problems the experiece of followig clear argumets ad of citig evidece to support their ow ideas a appreciatio of mathematics both as a ejoyable activity through which they experiece success, ad as a useful body of kowledge ad skills.
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY Outlie of the course This table outlies the course cotet uder each topic. For more detail refer to the Juior Certificate Mathematics syllabus ad the Juior Certificate Guidelies for Teachers: Mathematics. These are accessible o www.educatio.ie Topic Cotet details Sets Number systems 1. Listig of elemets of a set. Membership of a set defied by a rule. Uiverse, subsets. Null set (empty set). Equality of sets. 2. Ve diagrams. 3. Set operatios: itersectio ad uio (for two sets oly), complemet. 4. Commutative property for itersectio ad uio. 1. The set N of atural umbers. Order (<, <, >, >). Idea of place value. Sets of multiples. Lowest commo multiple. The operatios of additio, subtractio, multiplicatio ad divisio i N where the aswer is i N. Meaig of a for a, N, = 0. Evaluatio of expressios cotaiig at most oe level of brackets. Examples: 2 + 7 (4 1) 6 + 10 x 3 3(14 5) (7 + 2) Estimatio leadig to approximate aswers. 2. The set Z of itegers. Positioal order o the umber lie. The operatio of additio i Z. 3. The set Q + of positive ratioal umbers. Fractios: emphasis o fractios havig 2, 3, 4, 7, 8, 16, 5, 10, 100 ad 1000 as deomiators. Equivalet fractios. The operatios of additio, subtractio ad multiplicatio i Q +. Estimatio leadig to approximate aswers. Fractios expressed as decimals; for computatios without a calculator, computatio for fractios with the above deomiators excludig 3, 7 ad 16. Decimals: place value. The operatios of additio, subtractio, multiplicatio ad divisio. Roudig off to ot more tha three decimal places. Estimatio leadig to approximate aswers. Percetage: fractio to percetage. Suitable fractios ad decimals expressed as percetages. Example: 32 ; 32% 100 Equivalece of fractios, decimals ad percetages. Example: 42 ; 0.42; 42% 100 4. Squares ad square roots. 5. Commutative property. Priority of operatios.
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY Applied arithmetic ad measure 1. Bills: shoppig; electricity, telephoe, gas, etc. Value added tax (VAT). Applicatios to meter readigs ad to fixed ad variable charges. Percetage profit: to calculate sellig price whe give the cost price ad the percetage profit or loss; to calculate the percetage profit or loss whe give the cost ad sellig prices. Percetage discout. Compoud iterest for ot more tha three years. Calculatig icome tax. 2. SI uits of legth (m), area (m 2 ), volume (m 3 ), mass (kg), ad time (s). Multiples ad submultiples. Twety-four hour clock, trasport timetables. Relatioship betwee average speed, distace ad time. 3. Calculatig distace from a map. Use of scales o drawigs. Statistics ad data hadlig 4. Perimeter. Area: square, rectagle, triagle. Volume of rectagular solids (i.e. solids with uiform rectagular cross-sectio). Legth of circumferece of circle = π. Legth of diameter Use of formulae for legth of circumferece of circle (2πr), for area of disc (i.e. area of regio eclosed by circle, πr 2 ). Use of formula for volume of cylider (πr 2 h). 1. Collectig ad recordig data. Tabulatig data. Drawig ad iterpretig pictograms, bar-charts, pie-charts (agles to be multiples of 30 ad 45 ). Drawig ad iterpretig tred graphs. Relatioships expressed by sketchig such graphs ad by tables of data; iterpretatio of such sketches ad tables. 2. Discrete array expressed as a frequecy table. Mea ad mode.
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY Algebra 1. Formulae, idea of a ukow, idea of a variable (iformal treatmet). Evaluatio of expressios of forms such as ax + by ad a(x + y) where a, b, x, y N; evaluatio of quadratic expressios of the form x 2 + ax + b where a, b, x N. Examples: Fid the value of 3x + 7y ad of 6(x + y) for give values of x ad y. Fid the value of x 2 + 5x +7 whe x = 4. 2. Use of associative ad distributive properties to simplify expressios of forms such as: a(x ± b) + c(x ± d) x(x ± a) + b(x ± c) where a, b, c, d, x N. Examples: 3(x 2) + 2(x + 1) x(x + 1) + 2(x + 2) Relatios, fuctios ad graphs 3. Solutio of first degree equatios i oe variable where the solutio is a atural umber. Examples: Solve 3x + 4 = 19. Solve 4(x 1) = 12. 1. Couples. Use of arrow diagrams to illustrate relatios. Example: is greater tha 2. Plottig poits. Joiig poits to form a lie. 10 3. Drawig the graph of forms such as y = ax + b for a specified rage of values of x, where a, b N. Simple iterpretatio of the graph. Example: Draw the graph of y = 3x + 5 from x = 1 to x = 6.
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY Geometry 1. Sythetic geometry: Prelimiary cocepts: The plae. Lie ab, lie segmet [ab], ab as the legth of the lie segmet [ab]. Agle; amig a agle with three letters. Straight agle. Agle measure; abc as the measure of abc. Acute, right ad obtuse agles. Parallel lies; perpedicular lies. Vertically opposite agles. Triagle (scalee, isosceles, equilateral), quadrilateral (covex), parallelogram, rectagle, square. Practical, ituitive approach, for example usig drawigs ad paper-foldig. For costructios, the use of compasses, set squares, protractor, ad straight-edge are allowed uless otherwise specified. Use of geometrical istrumets ruler, compasses, set squares ad protractor to measure the legth of a give lie segmet, the size of a give agle ad the perimeter of a give square or rectagle. Costructio: To costruct a lie segmet of give legth (ruler allowed). Costructio: To costruct a triagle (ruler allowed) whe give: the legths of three sides; the legths of two sides ad the measure of the icluded agle; the legth of a base ad the measures of the base agles. 11 Fact : A straight agle measures 180. (For iterpretatio of the word Fact see Guidelies for Teachers.) Fact : Vertically opposite agles are equal i measure. Fact : The measure of the three agles of a triagle sum to 180. Costructio: To costruct a right-agled triagle, give sufficiet data (ruler allowed). Fact (Theorem of Pythagoras): I a right-agled triagle, the square of the legth of the side opposite to the right agle is equal to the sum of the squares of the legths of the other two sides (verificatio by fidig the areas of the squares o the three sides or otherwise). Costructio: To costruct a rectagle of give measuremets (ruler allowed). Fact : A diagoal bisects the area of a rectagle (verificatio by papercuttig or otherwise). Costructio: To draw a lie through a poit parallel to a give lie. Costructio: To divide a lie segmet ito two or three equal parts. Costructio: To bisect a agle without usig a protractor. Meaig of distace from a poit to a lie.
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY Geometry Meaig of base ad correspodig perpedicular height of a parallelogram. Fact : The area of a parallelogram = base x (correspodig) perpedicular height. 2. Trasformatio geometry: Cetral symmetry, axial symmetry. Use of istrumets to costruct the image (rectiliear figures oly) uder (i) axial symmetry ad (ii) cetral symmetry. 12
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY Usig Calculators The use of calculators is a regular feature i the daily mathematics lesso i post-primary schools. Calculators are a ivaluable aid for all studets ad especially those with mild learig disabilities. Most importatly, calculators eable studets with mild geeral learig difficulties to achieve success i mathematics. For some of these studets memorisig umber bods ca be a problem. This ca be offset by usig a calculator, as log as the uderlyig operatios ad cocepts are uderstood by the studets. The overall aim i the use of calculators i the mathematics class is to eable studets to kow whe it is appropriate to use metal methods, writte methods, ad calculator methods, or a combiatio of these. A simple drill of Thik first, write, use the calculator, write, ad the compare the results agaist the origial estimate should be practised. Roudig off ad writig the estimate before usig the calculator ca esure that cogitive skills are beig developed ad will avoid mechaical use of the calculator. Ecouragig estimatio Through skillful questioig studets ca be ecouraged to use estimatio. Iterestig questios such as, Are 150 hours more or less tha a week? will ecourage a estimated aswer followed by calculator work. Eterig correctly Eterig the decimal poit eeds careful attetio, especially i relatio to moey problems, for example 20 is etered differetly to 20c, ad i tur to 2c. The fractio symbol b Usig the fractio symbol o the calculator a c ca make fractio calculatio achievable for studets. 1 1 2 Oce mastered, a problem like 2 + 4 + 3 ca be aswered easily. Equivaleces ca be explored ad large fractios simplified. The teacher will eed to show this to the class ad to idividual studets. Studets workig i pairs or threes ca help each other. 13 The teacher will choose whether a scietific calculator or a simpler model will suit the studet. The latter usually has large keys ad is most effective i performig the four basic fuctios. However, i dealig with fractios the scietific models offer great relief from legthy ad complex algorithms. The calculator ca eable the studet to keep up with learig i the system of atural umbers, itegers, ratioal umbers, ad real umbers. The teacher will familiarise himelf/herself with the various brads of calculators beig used by the studets, ad assist them with mior differeces i operatig them. The followig are some approaches that teachers may fid useful. Pair ad small group work This ca be of great beefit whe studets are learig how to maipulate the calculator ad whe solvig problems. Iterpretig display correctly Pair work helps greatly with this. Questios like, What does 12.3 mea i a moey problem? How would 3 cets be represeted o the display? How would you key i twelve euro ad three cet? will stimulate a correct readig of the display. Studets should write their aswers o paper. I usig the memory butto a teacher ca judge the readiess of a studet for calculatios ivolvig more tha oe stage. For example, i calculatig the wall area of a room 250cm X 500cm X 300cm seems impossible for the studet with geeral learig difficulties, yet 2 X 3 X 5 whe stored to memory M+ (2 X 4 X 5 M+ ) ca be accessed by pressig Recall Memory. This is a algorithm that ca be practised ad used for larger umbers. Pair work is very useful i this cotext i developig cofidece ad i explorig methods.
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY Checkig aswers The calculator offers the teacher ways of showig the studet how to check aswers usig iverse operatios, thus reiforcig the learig of basic operatios. For example, 27-15 = 12 ca be checked by 12 + 15. Idetifyig studets miscoceptios ad errors By observig the studet ad by egagig with him/her i calculator work the teacher ca ascertai at what stage i the problem-solvig process a misuderstadig may have occurred, ad ca deal with it rather tha gettig bogged dow i calculatio processes. If a error occurs as a result of operatig the calculator icorrectly, it ca be dealt with o a idividual, group, or class basis. Teachig strategies Whe plaig for teachig ad learig i the area of Mathematical Studies ad Applicatios a variety of teachig strategies eeds to be cosidered. These will respod to the particular challeges studets with mild geeral learig disabilities experiece i egagig fully with, for example, mathematical laguage, oral ad writte commuicatio, problem-solvig, ad the retetio of facts ad cocepts. The tables that follow list some of the potetial areas of difficulty, ad suggest appropriate strategies for classroom use. It is importat to remember that ot all studets with mild geeral learig difficulties face all of these challeges. Neither is it a exhaustive list. These are the most commoly foud potetial areas of difficulty. Usig ICT May computer programs ca be used at differet levels withi oe group or class. Useful software is available commercially ad ca be ivaluable i reiforcig cocepts, i assistig calculator usage i computatio, i providig problem solvig opportuities ad i makig mathematics fu ad ejoyable. Studets workig, usig ICT, i groups or pairs ofte make sigificatly greater progress tha those who work idividually, ad this should be bore i mid i classroom plaig. The orgaisatio of ICT facilities varies eormously from school to school, yet the opportuity for studets with mild geeral learig difficulties to avail frequetly of ICT i the developmet of their mathematical skills is highly recommeded. 14
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY Addressig potetial areas of difficulty for studets with mild geeral learig disabilities s Potetial area of difficulty Short-term memory = Implicatios for learig Retetio of facts ad defiitios ca be a problem. + Possible strategies Ecourage the use of visual clues to aid memory, for example a verbal or writte defiitio of a isosceles triagle could be accompaied by a diagram with two sides ad two agles marked as equal. Ecourage studets to ivet rhymes, sogs or memoics to help them to recall facts ad practise estimatio skills, so that a calculator ca be used efficietly. Work o makig certai operatios automatic through usig fu games such as table-darts or fractio-decimal equivalece domioes. s Potetial area of difficulty Short attetio spa ad poor cocetratio = Implicatios for learig The studet fids it difficult to stay o a task, may rush the task, ad be easily distracted. + Possible strategies Provide shorter tasks with clear rewards for stayig o the task ad for completig it. For example, allow a studet to try a mathematical puzzle whe the task is completed. Use a variety of teachig methodologies; keep periods of istructio short ad to the poit, ad recap frequetly. Use teacher observatio efficietly ad ote achievemets, stregths, ad preferred learig styles i plaig future work. Ecourage studets to keep portfolios of work i which they record their mathematical achievemets. 15 s Potetial area of difficulty Uderstadig mathematical cocepts ad abstractios = Implicatios for learig The studet fids mathematics difficult ad has particular difficulty with certai abstract cocepts ecessary for algebra ad geometry. + Possible strategies Group discussio ca help the studet to liste to ad work with others. This is very useful whe itroducig a theme or cocept to promote a discussio with studets. I learig cocepts activities should be varied through the use of games, ICT, ad real-life problems relevat to the studet s experiece. Learig ca be made fu by usig fuy ames, silly scearios. or ulikely settigs. Ecourage co-operative learig activities, icludig pair-work ad small group exercises. Teach the topic i chuks rather tha as a sigle block.
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY s Potetial area of difficulty Spatial awareess = Implicatios for learig The studet may have difficulty orgaisig materials. She/he may display left/right cofusio whe recordig ad may ot recogise shapes if iverted or rotated. Topics such as geometry, applied arithmetic (area ad volume) ad graphig will be more challegig for this type of studet. + Possible strategies The studet should be give plety of work with three-dimesioal objects ad particular attetio should be paid to the laguage of spatial awareess. Ecourage the studet to make simple models (from cardboard or usig a dyamic geometry computer package) whe learig the properties of geometrical shapes. Whe discussig area or volume of a shape esure that the studet has access to a model of the relevat shape. Usig puzzles, tagrams, ad shape-makig kits i a fu way ca help the studet with this area of difficulty. Ecourage studets to be aware of their ow persoal space. Keep the patters of classroom orgaisatio cosistet. 16 s Potetial area of difficulty Applyig previously leared kowledge = Implicatios for learig The studet may fid it difficult to apply a skill or cocept already acquired i a differet settig, for example measurig i Geography, Sciece, or Home Ecoomics. + Possible strategies Revisit ad review previously leared kowledge regularly. Ecourage studets to revisit skills ad kowledge leared i a previous class, for example Yesterday we leared how to do a survey ad make a tally chart to show the results. Use a cross-curricular approach to the teachig of skills or cocepts that are commo to differet subjects, for example measurig agles i mathematics ad techical graphics or weighig i mathematics ad home ecoomics. Draw the studets attetio to what is happeig, for example This is just like the measurig we did last week. What did we use to measure our books? How did we place the ruler? Reiforce mathematical cocepts ecoutered i other areas of the curriculum ad ecourage the studet to make coectios, for example How did you fid the volume of somethig i sciece?
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY s Potetial area of difficulty Trasferrig of learig to real-life = Implicatios for learig The studet may ot use mathematics i real situatios. For example, she/he may ot use arithmetic whe buyig goods i a shop, may ot see the eed to measure whe cookig, or may ot recogise shapes i the eviromet. + Possible strategies Use real-life objects ad cois i appropriate situatios. Discuss with the studets how they sped moey. If possible, provide studets with opportuities to hadle real-life materials i real-life cotexts; for example, moey i a real shop or i the school shop. Esure that parets are aware of the importace of coutig ad hadlig moey or of measurig at home, for example sharig equally, weighig for cookig, or measurig whe doig DIY. s Potetial area of difficulty Visual sequecig + Possible strategies = Implicatios for learig The studet may ot be able to copy from the board or from a book ad may have difficulty with sequecig ad mirror writig. Teach the studet how to chuk iformatio ad how to check if it is correct; for example, copyig oly oe part of the sum at a time. Preset board work carefully, give clear istructios as to what the studet eeds to copy, ad use worksheets where appropriate. Use rhymes, sogs, memoics ad mid- maps to reiforce sequeces. Use visual cues. 17 s Potetial area of difficulty Cofusio with sigs ad symbols = Implicatios for learig The studet may ot read symbols ad may ask questios such as Is this a add sum? + Possible strategies Use charts to relate mathematical symbols to everyday symbols; for example, egg + chips, price, 20% off. Ecourage studets to verbalise what they are doig first; for example, lookig for ad idetifyig the symbol, or applyig the correct symbol if it is a writte problem. Ecourage studets to keep a symbol ad keyword dictioary.
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY s Potetial area of difficulty Laguage = Implicatios for learig The studet caot follow complex seteces or multiple meaigs ad may process oly part of the istructio. The studet fids it difficult to verbalise what she/he is doig i mathematics. The studet has difficulty i relatig the vocabulary of mathematics to real-life situatios. + Possible strategies Idetify specific mathematical terms ad esure that these are reiforced i differet settigs ad i other areas of experiece. Ecourage the studet to use relevat mathematical terms whe appropriate. Esure that the mathematical laguage used each week is commuicated clearly to studets ad their parets. Use a Key Word approach by displayig a wall chart of mathematical terms used each week, thus eablig the studet to build up a persoal mathematical dictioary. s Potetial area of difficulty Readig + Possible strategies = Implicatios for learig Readig difficulties ca prevet the studet from egagig with mathematics. He/she may be capable of completig the mathematical task but may become frustrated ad cofused by prited words. 18 Preset problems pictorially. Ask the studets to pick out the parts of the problem they ca read ad to focus o relevat iformatio. There is ofte a lot of redudat iformatio i a writte problem. Avoid presetig the studet with pages of textbook problems by givig modified worksheets (with diagrams) or verbally delivered istructios. s Potetial area of difficulty Followig istructios = Implicatios for learig The studet becomes cofused whe faced with more tha oe istructio at a time. + Possible strategies 00 Get the studet to repeat the istructio(s). Give short, clear istructios, ad use pictorial cues. Give verbal/writte hits. For example, use graph paper. What kid of problem is it? What do you eed to kow? What do you do ext? Preset clear guidace o how ad whe assistace will be give by the teacher/ other studets durig the lesso.
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY s Potetial area of difficulty Beig overwhelmed by the learig process = Implicatios for learig The studet becomes overwhelmed whe preseted with ew iformatio or skills ad cosequetly caot lear. + Possible strategies Adapt the materials give to a group. For example, have some compare the measure of agles usig a protractor while others use cut-out cards. Adapt teachig styles. For example, use more discussio at both the begiig ad ed of the lesso to help both teacher ad studet to uderstad how they are learig. Adapt the resposes required. The same activity ca ofte be doe with a group or class but some studets will aswer orally, some by usig symbolic represetatio, or some by usig a pictorial respose. Adapt the requiremets of the task. Oe group or idividual may oly have to do six of the questios, whereas aother may have to do te or more. Set persoal targets for the studets so that they do ot feel others are gettig less to do tha they are. 19
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY Exemplars These exemplars demostrate how certai strategies outlied i the previous sectio ca be used whe teachig a selectio of topics from the syllabus. These exemplars are ot iteded to cover the course or ay oe part of the course etirely. Teachers usig them are ecouraged to choose the learig outcomes, supportig activities, ad assessmet strategies that best suit the eeds of their studets. Some studets may oly achieve the first oe or two learig outcomes while others may achieve the full rage of outcomes. The importat factor is their iclusio i the experiece. The Juior Certificate Guidelies for Teachers: Mathematics cotais further exemplar material that ca be adapted to suit the eeds of a variety of studets. The guidelies also cotai useful refereces to mathematical resources ad websites. Structure of the exemplars Each of the exemplars is preceded by a outlie of the relevat sectios of the Primary School Curriculum, the Juior Certificate (foudatio/ordiary level) ad the Juior Certificate School Programme (JCSP). Some of the potetial difficulties experieced by studets with mild geeral learig disabilities that relate specifically to the area covered i the exemplar are outlied, ad suitable strategies are suggested. I additio, a approximate time scale, a list of resources, suggested outcomes, supportig activities, ad assessmet strategies for a lesso or series of lessos are provided. The exemplars are orgaised i the order i which the topics occur i the foudatio level mathematics syllabus. 20
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY No. Syllabus topic Exemplar Title Page 1 Mathematics Number systems Fractio attractio 22 2 Mathematics Applied arithmetic ad measure 3 Mathematics Applied arithmetic ad measure 4 Busiess Studies The Busiess of Livig Time What s the time 34 Walkig o the edge 45 Goig shoppig 54 Mathematics Applied Arithmetic ad Measure 5 Busiess Studies The Busiess of Livig Sources of icome ad iterpretig pay slips 63 6 Busiess Studies The Busiess of Livig Preparig Aalysed cash books 75 7 Busiess Studies Eterprise Statistics ad data hadlig 92 8 Mathematics Algebra Algebra activity 100 21 9 Mathematics Relatios, fuctios ad graphs Plottig poits 119 10 Mathematics Geometry What kid of triagle is it? 128 11 Busiess Studies The Busiess of Livig Icome ad expediture 135 12 Home Ecoomics Food Studies ad Culiary skills Desig ad make a pizza 139
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY Exemplar 1: Mathematics Syllabus topic: Mathematics: Number systems Fractio attractio Primary School Curriculum (5th ad 6th classes) Mathematics Strad: Number Strad uit: Fractios Juior Certificate (Ordiary level) Number systems: The set Q of ratioal umbers. Decimals, fractios, percetages. Decimals ad fractios plotted o the umber lie. The operatios of additio, subtractio, multiplicatio ad divisio i Q. Juior Certificate School Programme Use of umber: Apply the kowledge ad skills ecessary to perform mathematical calculatios Time scale: The full rage of learig ad assessmet activities preseted i this exemplar may take up to te class periods. Potetial areas of difficulty > Short term memory > Uderstadig cocepts > Spatial awareess > Trasfer to real-life > Vocabulary/laguage difficulties 22 Strategies used i this exemplar Specific targetig of mathematical laguage Applyig fractios to real-life situatios Usig reiforcemet techiques Usig games to reiforce cocepts ad operatios Usig cocrete materials ad maipulatives Usig cross-curricular liks to home ecoomics, materials techology, ad sciece. Usig pair ad group work i specified tasks
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY Exemplar 1: Mathematics Resources Bars of chocolate, pizzas or cardboard/plastic models that ca be easily divided Card, paper, material that ca be folded, tor, ad cut Sets of fractio strips Sets of fractio domioes Strig/wool, post-it otes, ad scissors Kitche weighig scales Soft driks cotaiers All activities are based o studets workig i pairs or groups of three or four. Activity 1 ca be coducted as a whole class discussio, as the teacher chooses. There is commercially available software o Fractios which is useful for reiforcig studets uderstadig of fractio cocepts, such as Display it Yourself Series: Fractios ad Maths made Easy. Suggested outcomes Supportig activities Assessmet strategies As a result of egagig i these activities studets should be eabled to 1. state where fractios are used i everyday life 1. Discuss where fractios occur i everyday life. 2. Sharig amog frieds 1. Studets draw a poster of a advertisemet or copy a recipe that shows the use of fractios i everyday situatios, ad give oral explaatios of the meaig of the fractios used i their posters. 23 2. split a variety of items ito differet fractios, usig fractio words ad terms appropriately 3. Foldig activities 2. Studets self-assess their work by askig questios such as: Was this sharig doe equally? What problems arose? 3. fold, tear, or cut a variety of items ito differet fractios, usig words ad terms appropriately. 3. Observe the studets discussig whether the foldig/tearig/ cuttig task bee accomplished ad what difficulties arose.
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY Exemplar 1: Mathematics Suggested outcomes Supportig activities Assessmet strategies 4. uderstad the cocept of fractios i terms of arragig them i order, ad as a way of describig how two thigs relate to each other 5. write fractios ad equivalet pairs of fractios 6. uderstad the eed for multiple ad LCM i additio list multiples 4. A series of group tasks usig real ad meaigful cotexts to explore fractio cocepts 5. Studets i pairs use couters to explore fractio relatioships 6. Discuss the eed for multiples, usig examples 4. Observe the studets egagig with tasks ad the extet to which they accomplish them. Ecourage self ad peer moitorig ad itervee whe ecessary. 5. Observe whether studets ca write fractios correctly o work-cards, usig umerator ad deomiator. 6. Observe whether the studets ca idetify the LCM 7. recogise equivalet fractios. 7. Play equivalet domioes games ad fractio bigo. explai the eed for LCM create lists of multiples ad select the lowest solve practical problems usig LCM. 24 7. Observe whether the studet matches equivalet fractios. Cross-curricular liks: These skills may be reiforced if similar cocepts i home ecoomics, materials techology, ad sciece are treated at the same time.
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY Exemplar 1: Mathematics Activity 1 Spot the fractio Usig appropriate visual aids, such as advertisemets ad labels, discuss where fractios occur i everyday life ad what expressios such as ½ price 1 3 extra free, add ¼ tsp of salt, half a metre, first half, half time, last quarter mea. Activity 2 Sharig with frieds The purpose of this activity is to give studets the opportuity to uderstad the cocept of simple fractios by sharig out items of various shapes amog various umbers of people. Studets experiece ad discuss sharig chocolate bars, pizzas, etc. with frieds. The activity ca be made simple (for example, sharig everythig betwee two people) or more complicated as appropriate. Resources Various bars of chocolate (A large 4x8 square bar of chocolate is a useful resource for this activity ad has eough squares to make a ice reward for everyoe at the ed of the lesso.) Note: A chocolate orage has 20 segmets ad so ca be used to demostrate the cocepts of halves, quarters, fifths, teths, ad twetieths. 25 A pizza ad pizza slicer Cardboard or plastic shapes that ca be used to represet bars of chocolate or pizzas whe it is ot possible or appropriate to use the real thig Laguage use Throughout the activity the teacher should liste to the laguage that the studets use to explai what they are doig. I additio to usig their ow words ad phrases studets should be ecouraged to uderstad ad use other appropriate keywords. Depedig o the actual activity, keywords may iclude share, split, slice, divide equally, how may pieces, what fractio, what is left, half, quarter, eighth, sixteeth. Skilled questioig by the teacher ca assist studets to move from the cocept of sharig or dividig to the cocept of fractio. Some sample sharig activities Studets may begi sharig usig the oe for you, oe for me method. Some studets may move o to workig out how may pieces there are, ad the how may each perso should get, without havig to deal the pieces out oe at a time. 1. A bar of chocolate has 6 squares, how would you (a) share it equally betwee 2 people? (b) share it equally betwee 3 people? 2. A bar of chocolate has 32 squares. How would you (a) share it equally betwee yourself ad a fried (2 people)? (b) share it equally betwee 4 people? (c) share it equally betwee 8 people? (d) share it equally betwee 16 people? (e) If you share it so that for every square you give a fried you get 3 squares, what fractio will your fried get ad what fractio will you get?
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY Exemplar 1: Mathematics 3. Sharig a pizza amog frieds. Use a real pizza or cardboard or plastic segmets. (a) 4 frieds wat to share a pizza.what fractio does each get? Show the size of slice that each will get. (b) 8 frieds wat to share a pizza.what fractio does each get? Show the size of slice that each will get. (c) 6 frieds order a pizza. Eoi ad Fraces are very hugry ad wat two slices each. Ae, Breda, Carmel ad Des wat oly oe slice each. How may slices should the pizza be cut ito? Show how much (ad what fractio) each will get. (d) A pizza is sliced ito 8 equal pieces. Áie has oe quarter, Sue has oe half ad Séamus has what is left. Show how may slices each perso gets. Activity 3 Foldig activities The purpose of this activity is to give studets the opportuity to uderstad the cocept of simple fractios by foldig, tearig, or cuttig items of various shapes. Laguage use As explaied i the previous activity studets should be ecouraged to use ad uderstad appropriate words, icludig fold, half, quarter, eighth, what fractio of, five out of eight, legth by breadth, legthways, square. Some sample activities Paper foldig of differet shapes 26 1. Foldig a rectagle ito quarters i differet ways: (a) Fold a rectagular (or square or circular) piece of material i half legthways ad cut alog fold. Do the same with the two pieces produced ad do the same agai. You should have 8 equal pieces of material. Measure the legth ad breadth of each piece of material. (b) Take a rectagular (or square or circular) piece of paper. Fold it i thirds (this may require some assistace), ad the i half. What fractio of the origial piece of paper is the folded piece? Ufold ad cout how may sectios have bee created. Colour i three of the sectios. What fractio of the paper is coloured? What fractio of the paper is ot coloured? Is there aother way of amig this fractio? (c) Similar coversatios ca be built aroud foldig shapes split ito differet fractios i differet ways. (d) Note that this paper-foldig method is useful whe teachig multiplicatio of fractios. For example (b) above demostrates that half of a third is a sixth. Cuttig sectios of material for a patchwork quilt. Take a piece of material measurig 32cm by 16cm. Fold material i half legthways ad cut alog the fold. Do the same with the two pieces produced ad do the same agai. You should have 8 squares of material. Measure the legth ad breadth of each piece of material (8cm by 8cm). By likig with the Home Ecoomics teacher it might be possible to make a patchwork quilt ad studets could see what fractio of the quilt is made up from their colour or patter of material.
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY Exemplar 1: Mathematics Activity 4: Group Activities (oe-two class periods) Measurig ad recordig fractios i differet cotexts Studets will work i groups, movig o the completio of oe task to the ext, but stayig util a task is completed. Co-operative learig is the key approach here. Task 1: How log is a piece of strig? Studets are required to measure ad label strig/wool legths as directed o task cards, for example 1 3m, ¼m, 300 1000m, 2 5m, 3 10m, 40 100m, 1 10m, ½m, 1 3, ½m, 1 5m, etc. Studets place these o a wall-chart, appropriately labeled. They arrage the strigs i ascedig/descedig order, writig the list of fractios udereath. Pieces of the same legth are recorded i a sectio called equivalet fractios, for example ¼m = 250 1000. Calculators ca be used to simplify fractios ad to assist i explorig equivaleces. Task 2: The weight of kowledge Testig ad recordig equivaleces ad relatioships. Weigh schoolbooks usig kitche style scales. Record the weights o a wall chart, for example: Maths book : 400g A4 copy : 200g Write dow relatioships, for example a Maths copy weighs 200/400 or 1/2 the weight of a book. 27 This could be made ito a fu exercise as follows: Add all weights ad see how much you carry every day. If you were allowed leave your copies at home what fractio would that be of your bag s total weight? What fractio do your books make? Weigh the books you like. What fractio of the total are they? Record all these. Allow studets to record this iformatio i their Maths copies or record books. They will probably wat to do this ad brig it home to discuss it with their parets. Task 3: Thirsty work Comparig volume of soft drik bottles, cas, ad cartos. Usig 2l, 1.5l, 1l, 500ml, 330ml, 250ml ad 200ml bottles make out fractio relatioships (placig oe umber over the lie ad the other umber uder the lie). Ask the studets to cosider how may 500ml bottles would give the same amout as the 2l bottle? So, a 500ml bottle is a?/? of the 2000ml bottle? 500ml bottle = 500 = 1 2000ml bottle 2000 4 Calculators should be used to check simplified fractios. Other bottles, cas ad cartos ca be dealt with similarly. This could be exteded to iclude prices ad value for moey. Prices could be compared also usig fractios.
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY Exemplar 1: Mathematics Activity 5 Workig with couters ad fillig work-cards Discuss with the studets that, for example, oe third meas dividig ito three parts. Ecourage them to talk through what they are doig. For example, Group 1 could use 12 couters/items ad complete cards such as 1 = 2 = 3 12 3 12 1 = 1 = 2 = 3 = 2 12 4 12 4 12 4 12 Based o16 ad 20 couters (ad other umbers) similar workcards ca be made out ad studets ca explore ad record equivaleces. Activity 6 Learig multiples Explai or discuss the eed for multiples, usig a simple example: 28 ¼ ¼ ¼ ½ ½ + = 2 4 + = 3 4 ¼ ¼ ¼ ¼ The studets list the multiples of 2 ad write them i their copies. This is repeated with the multiples of 4. They circle the lowest oe that is commo to both lists ad write: LCM = This ca be repeated for other pairs of umbers. Some studets might have a problem because they do ot kow their multiplicatio bods. The teacher ca work o that with those studets, or ecourage them to use a calculator i completig their lists. Whe learig multiples the studets list the multiples of various umbers.
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY Exemplar 1: Mathematics Workig i pairs or groups of three the studets ca complete the followig lists. Calculators ca be used Multiples of 2 = 2,, 6, 8,,,,,, 20,,, Multiples of 3 = 3,, 6,, 12,,, 21,, 27,,, Multiples of 4 = 4, 8,,, 20,,, 32,,. Multiples of 5 =,, 15, 20,, 30,,,, 50. Multiples of 6 =, 12,,, 30,, 42,, 54, 60. Multiples of 7 = 7,,, 28, 35, 42,,,, 70. Multiples of 8 =, 16,,, 40,,, 64,,. Multiples of 9 =,,,, 45, 54, 63,,, 90. Still i pairs, studets take turs highlightig/pickig out the lowest commo multiple (LCM) i two rows, completig workcards such as the followig : 29 Write the LCM of the followig umbers LCM of 2 ad 3 is LCM of 2, 3 ad 4 is LCM of 5 ad 7 is LCM of 8 ad 6 is
Guidelies Mild Geeral Learig Disabilities / Mathematics / POST-PRIMARY Exemplar 1: Mathematics Activity 7 Fractio games Resources Each studet playig the game eeds a set of fractioal parts. Each set of cards should be marked with a differet iitial (A, B, C, etc.) so that the sets ca be re-formed at the ed of the game. The game ca be played by betwee two ad four studets. 1. Cut from light-weight cardboard five rectagular sectios measurig 24cm by 6cm. 2. Mark oe strip as a uit measure. 1 3. Mark the other four strips ito fractioal parts as show below. Cut alog the lies, mark each piece with the same iitial, ad place the pieces ito a evelope for storage. ½ ¼ ¼ 30 1 3 1 3 1 6 1 6 1 6 1 6 1 6 1 6 1 12 1 12 1 12 1 12 ¼ 1 8 1 8 1 8 1 8 1 8 1 8 4. The fractioal pieces ca be made more robust by lamiatig them. 5. A umber of differet games are possible usig these fractioal pieces. Two games are described below. A. Make oe! 1. Each player is give a evelope of fractioal parts as described above. Each player retais his/her uit strip as a measure. All players empty the rest of their pieces ito a commo pile ad the pieces are mixed up. 2. Each player i tur draws a piece from the pile util all the pieces are goe. 3. Each player the assembles his/her pieces to make as may uit strips as possible. 4. The first player to costruct three uit strips (or whole) is the wier. B. Make half! 1. The game begis as above, but a time limit of three miutes is set. 2. The player who ca make the greatest umber of strips equivalet to a half is the wier. Other activities As a iitial itroductio to dealig with fractioal parts, studets ca be ecouraged to solve real-life problems based o splittig a pizza or a bar of chocolate ito various fractioal parts. Paper foldig activities such as those outlied i the Primary School Curriculum: Mathematics Teacher Guidelies are also useful.