Multiplicative Change Of - QUT
XLR8 – Accelerating Mathematics LearningXLR8 Unit 0303Multiplicativechange ofquantities2016Prepared by YuMi Deadly CentreFaculty of Education, QUT, Kelvin Grove
ACKNOWLEDGEMENTSWe acknowledge the traditional owners and custodians of the lands in which themathematics ideas for this resource were developed, refined and presented inprofessional development sessions.YUMI DEADLY CENTREThe YuMi Deadly Centre is a Research Centre within the Faculty of Education at QueenslandUniversity of Technology which aims to improve the mathematics learning, employment and lifechances of Aboriginal and Torres Strait Islander and low socio-economic status students at earlychildhood, primary and secondary levels, in vocational education and training courses, and through afocus on community within schools and neighbourhoods.“YuMi” is a Torres Strait Islander Creole word meaning “you and me” but is used here withpermission from the Torres Strait Islanders’ Regional Education Council to mean working together asa community for the betterment of education for all. “Deadly” is an Aboriginal word used widelyacross Australia to mean smart in terms of being the best one can be in learning and life.YuMi Deadly Centre’s motif was developed by Blacklines to depict learning, empowerment, andgrowth within country/community. The three key elements are the individual (represented by theinner seed), the community (represented by the leaf), and the journey/pathway of learning(represented by the curved line which winds around and up through the leaf). As such, the motifillustrates the YuMi Deadly Centre’s vision: Growing community through education.More information about the YuMi Deadly Centre can be found at http://ydc.qut.edu.au and staff canbe contacted at ydc@qut.edu.au.CONDITIONS OF USE AND RESTRICTED WAIVER OF COPYRIGHTCopyright and all other intellectual property rights in relation to this booklet (the Work) are ownedby the Queensland University of Technology (QUT). Except under the conditions of the restrictedwaiver of copyright below, no part of the Work may be reproduced or otherwise used for anypurpose without receiving the prior written consent of QUT to do so.The Work may only be used by schools that have received professional development as part of theAccelerating mathematics learning (XLR8) project. The Work is subject to a restricted waiver ofcopyright to allow copies to be made within the XLR8 project, subject to the following conditions:1.all copies shall be made without alteration or abridgement and must retain acknowledgement ofthe copyright;2.the Work must not be copied for the purposes of sale or hire or otherwise be used to deriverevenue;3.the restricted waiver of copyright is not transferable and may be withdrawn if any of theseconditions are breached. QUT YuMi Deadly Centre 2016Page iiUnit 03: Multiplicative change of quantitiesYuMi Deadly Maths
ContentsXLR8 Program: Scope and Sequence . ivOverview .1Context . 1Scope . 1Assessment . 1Cycle Sequence . 3Literacy Development. 4Can you do this? #1 .1Cycle 1: Repeated Additive Change .6Overview. 6RAMR Cycle . 8Can you do this? #2 . 15Cycle 2: Combining Equal Groups . 18Overview. 18RAMR Cycle . 20Can you do this? #3 . 25Cycle 3: Sharing and Separating Collections . 28Overview. 28RAMR Cycle . 30Can you do this? #4 . 35Cycle 4: Factorisation . 36Overview. 36RAMR Cycle . 38Can you do this? #5 . 43Cycle 5: Multiplicative Comparison . 46Overview. 46RAMR Cycle . 48Can you do this? #6 . 51Cycle 6: Multiplicative Combinations . 54Overview. 54RAMR Cycle . 56Can you do this? #7 . 61Cycle 7: Multiplicative Strategies for Larger Numbers . 62Overview. 62RAMR Cycle . 64Unit 03 Portfolio Task – Teacher Guide . 68The Big Party .1Can you do this now? Unit 03 .1List of FiguresFigure 1 Scope of this unit .2XLR8-ARC Booklets: QUT YuMi Deadly Centre 2016Unit 03: Multiplicative change of quantities Page iii
XLR8 Program: Scope and Sequence2 yearprogram3 yearprogramUnit 01: Comparing, counting and representing quantityStudents study countable attributes of their immediate environment, includingattributes of the group of students (e.g., more boys than girls, less students with blueeyes than brown eyes) in the classroom and attributes of the school (e.g., quantity ofwindows in a classroom, quantity of ceiling tiles, length of classroom in steps). Thiscontext is limited to those attributes which can be described and represented usingwhole numbers.11Unit 02: Additive change of quantitiesStudents extend their investigations of numbers from features of their immediateenvironment, to features of larger populations in their state, country or world. Thiscontext is limited to those features which can be counted using whole numbers andwhich can be used in additive number stories (for which the total or one of the parts isunknown).11Unit 03: Multiplicative change of quantitiesStudents explore multiplicative relationships and changes using real-world situationsthat involve discrete items. This context is limited to those features which can becounted using whole numbers, can be used in multiplicative number stories (for whichthe product or one of the factors is unknown), and for which divisions also result inwhole numbers.11Unit 04: Investigating, measuring and changing shapesStudents explore 3D objects, their 2D surfaces and the 1D attributes of point, line andangle. This context includes measurement of the attribute of turn (angle) andmathematical transformations of 2D shapes and 3D objects including reflection, rotationand translation of shapes and how these may be combined with tessellation to generateand describe designs.11Unit 05: Dealing with remaindersStudents extend their investigations of partitioning and quotitioning features of theirimmediate environment and features of larger populations in their state, country orworld to include situations that result in a remainder. Students will explore partitioningwhole items into fractions, quotitioning into smaller units, and sharing of remainders ofcollections. Continuous measures such as length provide useful contexts for partitioningand quotitioning.11Unit 06: Operations with fractions and decimalsStudents connect the common fraction representations of tenths, hundredths andthousandths to their decimal fraction representations in contexts common to students’immediate environments including money, measurement and parts of discrete wholes.Students will develop strategies to calculate additive and multiplicative changesinvolving fractional amounts represented as both common fractions and decimals.12Unit 07: PercentagesStudents extend their representations of fractions to include percentage. Percentage isused to compare values multiplicatively and to describe quantity comparisons,recommended daily intake of nutrients, discounts, markups, tax and simple interest.Students will be encouraged to work flexibly between common fractions, decimalfractions and percentages.12Page ivYuMi Deadly MathsUnit 03: Multiplicative change of quantities
2 yearprogram3 yearprogramUnit 08: Calculating coverageStudents extend their investigations of attribute measurement from one-dimensionallength measures to two-dimensional measures of coverage or area. This idea starts withcoverage which can be counted using whole numbers before extending to fractionalmeasures. Area measurement and calculation provides an opportunity for consolidationof multiplication and division with larger numbers, and consolidation of multiplicationand division of fractional quantities expressed as common fractions, mixed numbers ordecimal numbers.22Unit 09: Measuring and maintaining ratios of quantitiesStudents develop their ability to measure duration, convert between units of measureand describe proportional relationships between quantities of discrete items ormeasurements using ratio notation. Students will also explore changing overallquantities while maintaining consistent proportions between the parts.22Unit 10: Summarising data with statisticsStudents develop their ability to gather, organise and represent data from primary andsecondary sources. Ideas of sample, population and inference will be used to informdecision making from the gathered data. Students will also develop their ability toanalyse measures of central tendency and variation within data sets and learn torepresent and interpret these aspects on graphical representations (stem and leaf plotsand box and whiskers graphs). Further analysis of the misrepresentation of data willconclude this unit’s development of ideas surrounding critical analysis and interpretationof data and statistics.22Unit 11: Describing location and movementStudents develop their ability to describe location and movement along a 1D line and in2D space with respect to an origin and extending from internal to external frames ofreference. Generating 2D representations of location and movement on scale maps andgrids using alphanumeric coordinates and compass bearings and distance will beextended to include geometric location of points and collections of points on theCartesian plane. Students will explore Pythagoras’ theorem to find diagonal distancestravelled.23Unit 12: Enlarging maps and plansStudents develop their ability to describe proportional relationships between quantitiesof measurements using ratio notation. Ratio will also be used to describe enlargementand reduction transformations to create similar shapes, scale maps and grids,representations of shapes and paths on the Cartesian plane, and plan drawings.Explorations can be extended to trigonometric ratios between similar figures and theapplication of scale factor to area of similar figures.23Unit 13: Modelling with linear relationshipsStudents explore parallels between ratio and rate in the context of relationshipsbetween measured attributes. These understandings will be extended to algebraicequations which can also be represented on the Cartesian plane to assist withvisualisation of relationships and use of equations and algebraic calculations for findinggradient and distances between points on a line.23Unit 14: Volume of 3D objectsStudents explore relationships between measurements of solid objects that lead tocalculations of formulae, relationships between solid volume and surface area andinvestigations of contexts that require calculation of solid volume of composite objects.23Unit 15: Extended probabilityStudents extend upon their ability to determine theoretical probability and makeinferences based upon likelihood of an event. Students will explore and comparetheoretical and experimental probabilities, recognise when events are mutuallyinclusive, mutually exclusive or complementary and determine the probability of singlestep and multi-step events.23XLR8-ARC Booklets: QUT YuMi Deadly Centre 2016Unit 03: Multiplicative change of quantities Page v
OverviewContextIn this unit, students will explore multiplicative relationships and changes using real-world situationsthat involve discrete items. This context is limited to those features which can be counted usingwhole numbers, which can be used in multiplicative number stories (for which the product or one ofthe factors is unknown), and for which divisions also result in whole numbers.ScopeThis unit is based upon the number-as-count meaning of cardinal number. Once a collection iscounted, the collection can be arranged using simple multiplicative relationships that describe thecollection in terms of the product of factors. These relationships can be referred to as factor-factorproduct. Multiplication strategies can be used to compute the product if the factors are known.Division strategies can be used to compute the unknown factor if the other factor and the productare known.These relationships can be applied to a range of contexts including scaling of axes on graphs using amultiplier, and partitioning measured lengths to find how many tiles are needed to span a specifiedlength.The organisation of these and other related concepts is shown in Figure 1, in which the scope ofconcepts to be developed in this unit is highlighted in blue, concepts that may be connected to andreinforced are highlighted in green and number and algebra concepts and processes that arereinforced and applied within this area are highlighted in black.AssessmentThis unit provides a variety of items that may be considered as evidence of students’ demonstrationof learning outcomes: Diagnostic Worksheets: The diagnostic worksheet should be completed before starting to teacheach RAMR cycle. This may show what students already understand. Not all objectives arerepresented on diagnostic worksheets. Anecdotal Evidence: Some evidence of student understanding is best gathered throughobservation or questions. A checklist may be used to record these instances. Summative Worksheet: The summative worksheet should be completed at the end of teachingthe unit. This may be compared with student achievement on the diagnostic worksheets todetermine student improvement in understanding. Portfolio task: The portfolio task P3: The Big Party accompanying Unit 03 engages students withexploring multiplicative operations in the context of planning catering and seating for a largeparty.This task could be further extended by providing students with access to actual shoppingcatalogues and allowing them to make their own choices for party food and drinks although forthis unit we have limited computation to whole number values only.XLR8-ARC Booklets: QUT YuMi Deadly Centre 2016Unit 03: Multiplicative change of quantities Page 1
Figure 1 Scope of this unitPage 2 OverviewYuMi Deadly MathsSymmetryVolume2D shape 1D lineEntityNumberlineMovementPerimeter3D iptiveOrder eticexpressionAlgebraic rRepresentationDesignEquality ntFractionPlace gCongruenceStatisticsSamplespaceLikelihood
Cycle SequenceIn this unit, concepts identified in the preceding section are developed in the following suggestedsequence:Cycle 1: Repeated Additive ChangeIn this cycle, the simplest meanings for multiplication and division of repeated addition and repeatedsubtraction of multiple same-size sets are explored using a series of function machines with the samechange on each machine. The function machine is also used to explore multiplicative inverse,backtracking and recording of multiplicative change. Equivalence between additive and multiplicativeequations is explored using physical and drawn balances.Cycle 2: Combining Equal GroupsIn this cycle, activities extend the repeated addition meaning for multiplication to the combiningequal groups meaning (product meaning), using set and array model representations. This cycle willalso develop basic fact strategies built around field principles of commutativity, distributivity,associativity, identity and inverse.Cycle 3: Sharing and Separating CollectionsIn this cycle, the partitioning meaning of division is developed and identified as an inverse ofcombining. This cycle will also develop the symbolic notation for division, basic fact strategies, andconsolidate students’ interpretation and construction of worded multiplicative problems.Cycle 4: FactorisationA facility to work flexibly with quantities is an essential component of mental and writtencomputation strategies, and is also used to determine equivalence in fractions and ratios.Factorisation both relies on, and reinforces number facts as it focuses on equivalent numbersentences. This idea extends from factorisation of numeric quantities to simplification and expansionof expressions containing variables. This cycle also provides an opportunity to explore and discussodd, even, prime, composite and square numbers.Cycle 5: Multiplicative ComparisonIn this cycle, activities broaden the range of meanings for multiplicative relationships to include staticcomparison of groups where no change occurs (e.g., 3 times as many, 5 times fewer).Cycle 6: Multiplicative CombinationsIn this cycle, activities extend the meanings for multiplicative relationships to include multiplicativecombinations (e.g., 3 bread types, 5 possible fillings, how many possible sandwiches on the menu).Cycle 7: Multiplicative Strategies for Larger NumbersTraditionally, calculation of multiples of greater than single digit numbers needed to be completedmentally or manually as did division calculations (now often completed with calculators). Arguably,strategies for written and mental calculation are important for instances where calculating devicesare not available. However, mental and written calculation strategies also develop and demonstratea facility to work flexibly with number, enhance logical thinking processes, promote problem solvingstrategies, build number sense and are generalisable to strategies that extend to algebraicunderstandings and field properties.Notes on Cycle Sequence:The proposed cycle sequence outlined should be completed sequentially as it is presented.XLR8-ARC Booklets: QUT YuMi Deadly Centre 2016Unit 03: Multiplicative change of quantities Page 3
Literacy DevelopmentCore to the development of number and operation concepts and their expression at varying levels ofrepresentational abstraction (from concrete-enactive through to symbolic) is the use of languagethat is consistent with the organisation of the mathematical concepts. In this unit the following keylanguage should be explicitly developed with students, ensuring that students understand both theeveryday and mathematical uses of each term and, where applicable, the differences and similaritiesbetween these.Cycle 1: Repeated Additive ChangeRepeated addition, groups of, lots of, same size groups, grouping, multiples, times, multiplication,repeated subtraction, sharing, shared, partitioned, division, divided, undoes, opposite, inverse, input,output, change, balance, same as, equivalence, equivalentCycle 2: Combining Equal GroupsGroups of, lots of, sets of, array, multiple, factor, product, multiply multiplication, combining equalgroups, turnaround, identity, commutative law, associative law, distributive law, unknown, variableCycle 3: Sharing and Separating CollectionsDivision, shared, partitioned, divided, sets, groups, rows, columns, equal shares, inverseCycle 4: FactorisationGroups, sets, collections, array, multiple, factor, product, multiply, multiplication, divide, division,compare, comparison, inverse, unknown, variableCycle 5: Multiplicative ComparisonGroups, sets, collections, array, multiple, factor, product, multiply, multiplication, divide, division,compare, comparison, inverse, unknown, variableCycle 6: Multiplicative CombinationsMultiple, factor, product, multiply, multiplication, divide, division, compare, combinations, options,possibilities, inverse, unknown, variableCycle 7: Multiplicative Strategies for Larger NumbersMultiplication, division, array, area model, number fact strategies, factors, multiples, brackets,strategies, mental computation, separation, sequencing, compensationPage 4 OverviewYuMi Deadly Maths
Name:Date:Can you do this? #11.Match the stories to the pictures and symbols that represent them.PicturesStoriesSymbolsa) A relay team had 4 people running.Each person ran 3 laps of the oval.There were 12 laps run altogether.4 3 12b) Three children had 3 ballons each.There were 9 balloons altogether.6 2 12c) Twelve students entered theclassroom in pairs. Six pairs enteredthe room.3 3 9Obj.3.1.1a)i ii b)i ii c)i ii 2. Fill in the blanks in the Input/Output Tables from a Function 641274251595471127243. Draw a multiplication backtracking diagram for 4 4 4 12.4. Is the following equation true or false? 1 XLR8-ARC Booklets: QUT YuMi Deadly Centre 2016Obj.3.1.4a)i. ii. iii. b)i. ii. iii. T F Obj.3.1.5i. ii. iii. Obj.3.1.2iv. Obj.3.1.3 Unit 03: Cycle 1: Can you do this?
5. For each of the following stories:i. Underline the factors.ii. Circle the product.iii. Write an equation for the story with a symbol for the unknown.iv. Write down the answer(a) 21 students gave 6 to the class trip. How much money did theclass have altogether?(b) I bought 3 phones for 60 each. How much did I pay in total?Obj.3.1.6a)i. b)i. Obj.3.1.7a)ii. b)ii. Obj.3.1.8a)iii. b)iii. Obj.3.1.9a)iv. b)iv. 6. A paint store works out the price of its tins of paints using theequation: Q 5 3 C(Q is the quantity of paint ordered in Litres; C is the final price)If you order 8L of paint, what will be the price of the paint?Unit 03: Cycle 1: Can you do this?Obj.3.1.10i. ii. iii. YuMi Deadly Maths
Cycle 1: Repeated Additive ChangeOverviewBig IdeaThe simplest meanings for multiplication and division are repeated addition and repeatedsubtraction, where multiple, same-size sets are either added together (multiplication) or subtracted(division) a number of times. This idea can be explored with a series of function machines with thesame change on each machine. As for addition and subtraction, the function machine can also beused to explore multiplicative inverse, backtracking and methods of recording multiplicative changeas equations. It is also beneficial to reinforce the meaning of the equals sign as “same value as”.ObjectivesBy the end of this cycle, students should be able to:3.1.1 Act out, interpret and represent repeated addition multiplication stories informally. [2NA031]3.1.2 Explore multiplicative inverse for multiplication using function machines. [2NA032]3.1.3 Identify identity element for multiplication. [5NA098]3.1.4 Record multiplicative change and inverse using input-output tables. [4NA081]3.1.5 Record multiplicative change and inverse using backtracking diagrams. [5NA121]3.1.6 Identify factors within repeated addition multiplication stories as number of repeats and sizeof repeats. [5NA098]3.1.7 Identify the product within repeated addition multiplication stories as the total to be found.[5NA098]3.1.8 Represent repeated addition multiplication stories as equations using symbols for unknowns.[7NA175]3.1.9 Solve repeated addition multiplication stories using equations with a symbol for the unknown.[4NA082]3.1.10Evaluate algebraic expressions by substituting a given value for each variable. [7NA176]Conceptual LinksThe conceptual understanding from this cycle is necessary for understanding and expressingmultiplicative operations as written equations for the remaining cycles in this unit.MaterialsFor Cycle 1 you may need: Variety of everyday items to use to demonstrate sameness or equivalence Simple pan balancePage 6 Cycle 1: Repeated additive changeFunction machine (repeatable)Input, output and change cardsYuMi Deadly Maths
Key LanguageRepeated addition, groups of, lots of, same-size groups, grouping, multiples, times, multiplication,repeated subtraction, sharing, shared, partitioned, division, divided, undoes, opposite, inverse, input,output, change, balance, same as, equivalence, equivalentDefinitionsPartition: separate into a number of same-size pieces or groups, usually by sharing.Repeated addition: same number added a quantity of times. Most easily represented using set orlength materials. For example, 4 2 how many are 4 groups of 2 2 2 2 2 8.Repeated subtraction: same number subtracted a quantity of times. Most easily represented usingset or length materials. For example, 8 2 is the same as how many times can you take agroup of 2 from 8 8 – 2 – 2 – 2 – 2 4 times.AssessmentAnecdotal EvidenceSome possible prompting questions: How can you find how many there are altogether? Is there a simpler way to say this instead of using addition? Are the groups you have all the same size? How many groups do you have? Can you write an equation for this story using multiplication?Portfolio TaskThe student portfolio task P3: The Big Party provides students with opportunities to practisemultiplication as repeated addition.XLR8-ARC Booklets: QUT YuMi Deadly Centre 2016Unit 03: Multiplicative change of quantities Page 7
RAMR CycleThe focus of this cycle is to understand the conceptual difference between additive changes whichtake the form of part-part-total, and multiplicative changes which take the form of factor-factorproduct. It is also important to consolidate equivalence and the role of equals in regards to theconcept of multiplicative change. Using the meaning of multiplicative change as repeated additive
Unit 03: Multiplicative change of quantities Students explore multiplicative relationships and changes using realworld situations - that involve discrete items. This context is limited to those features which can be counted using whole numbers, can be used in
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across Brisbane, coastal and regional Queensland. In early 2017, Burger Urge had 20 burger bars, with more scheduled to open. Burger Urge operates both company owned and franchised restaurants. The feature article from Autumn 2013 edition of QUT alumni magazine QUT Links capture the early years of Burger Urge business story:
Jun 04, 2017 · Students’ completed copies of Student Activity Book, p. 15A or C2 (from Activity 2) session FolloW-uP Daily Practice Student Activity Book, p. 15B or c3, More Multiplicative comparison Problems Make copies. (as needed) Student Math Handbook, pp. 29–34 session 1.6A Multiplicative comparison cc
statements of multiplicative comparisons as multiplication equations. 4.OA.2 Use the four operations with whole numbers to solve problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative compa
Represent verbal statements of multiplicative comparisons as multiplication equations. 2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparis
Additive inverse and multiplicative inverse; the sum of additive inverses equals ; the product of multiplicative inverses equals . . Find the sum of and the opposite of . Write an equivalent expression using the fewest number of terms. Justify each step. 2. For and the multiplicative inver
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