· ... (length,(width,(product,(%mes ... (24,(amul%ple(of(2,(amul%ple(of(3,(amul%ple(of(4,(amul...

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Australian Curriculum Year 5 ACMNA098 Iden%fy and describe factors and mul%ples of whole numbers and use them to solve problems Key Idea Each coun%ng number is divisible by one and itself, and some coun%ng numbers are also divisible by other numbers Mul%plica%on which has an inverse rela%onship with with division, is helpful in finding factors Resources FISH Problem Solving kit Closing the Gap – Ladybirds Cubes / MAB units/Games h#p://nzmaths.co.nz/resource/mul4plesand factors ) cardboard squares mul%plica%on table (for differen%ated learners) poster size s%cky notes (for living charts) or alterna%vely use cardboard IWB Promethean Planet sample flipchart images displayed below for further clarifica%on. Google images (some aRached below) this is and individual learning. Vocabulary factors, mul%ples, prime number, divide, mul%ply, rectangle, length, width, product, %mes, lowest common, highest common Ac>vity Process Introductory Ac4vity Process The purpose of this lesson is to introduce a revision finding a factor and using mul%ples. (This will demonstrate what the students know and will determine how to proceed for the lessons ahead). Discuss the defini%ons of mul%ples and factors before your students begin this ac%vity. Make sure that they understand that every number is a factor of itself, because if they divide a number by itself, there is no remainder. For example, 12 ÷ 12 = 1 without a remainder, so 12 is a factor of 12. A prime number is a number that has only two factors, itself and 1, for example: 5, 7, 13, and 29. (Note that 1 itself is not considered to be a prime number.) Before the students play the game, ask the following ques%ons: • Imagine you threw a 4 and a 6. Which squares could you choose to cover with your counter?(a number with more than two factors, a factor of 24, a mul%ple of 2, a mul%ple of 3, a mul%ple of 4, a mul%ple of 8, an even number, or a mul%ple of 6) • Imagine you need a mul%ple of 5 to get four counters in a row. Which throws of the dice would give you a mul%ple of 5? (1 and 5, 2 and 5, 3 and 5, 4 and 5, 5 and 5, 6 and 5, 7 and 5, 8 and 5, or 9 and 5) (Adapted from h#p://nzmaths.co.nz /resource/mul4plesandfactors ) Teacher background informa6on to be shared and discussed or for students to record into a maths journal/dic6onary: The factors of a number are any numbers that divide into it exactly. This includes 1 and the number itself. For example, the factors of 6 are 1, 2, 3 and 6. The factors of 8 are 1, 2, 4 and 8. For larger numbers it is some%mes easier to 'pair' the factors by wri%ng them as mul%plica%ons. For example, 24 = 1 x 24 = 2 x 12 = 3 x 8 = 4 x 6 So the factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. Learners complete 3 factor and mul%ple examples eg 2 is a factor of 8, 8 is a mul%ple of 2 Ac4vity ProcessGame On This game could be extended by asking: • What are all the different products you could throw with the two game dice, one labelled 1–6 and the other 4–9? (4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 54) • There are two different ways of gegng a product of 12: throwing a 3 and a 4 or a 2 and a 6. Which other products can you throw more than one way using the game dice? (8: 1 x 8 or 2 x 4; 16: 2 x 8 or 4 x 4; 18: 2 x

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Page 1:  · ... (length,(width,(product,(%mes ... (24,(amul%ple(of(2,(amul%ple(of(3,(amul%ple(of(4,(amul ... appropriate(units(of(measurementfor(length,(area,(volume

Australian  Curriculum  Year  5  ACMNA098  Iden%fy  and  describe  factors  and  mul%ples  of  whole  numbers  and  use  them  to  solve  problems    

Key  Idea  •  Each  coun%ng  number  is  divisible  by  one  and  

itself,  and  some  coun%ng  numbers  are  also  divisible  by  other  numbers  

•  Mul%plica%on  which  has  an  inverse  rela%onship  with  with  division,  is  helpful  in  finding  factors  

 Resources    FISH  Problem  Solving  kit  Closing  the  Gap  –  Ladybirds  Cubes  /  MAB  units/Games  h#p://nzmaths.co.nz/resource/mul4ples-­‐and-­‐factors)  cardboard  squares  mul%plica%on  table  (for  differen%ated  learners)  poster  size  s%cky  notes  (for  living  charts)  or  alterna%vely  use  cardboard  IWB  -­‐  Promethean  Planet  -­‐  sample  flipchart  images  displayed  below  for  further  clarifica%on.  Google  images  (some  aRached  below)  -­‐  this  is  and  individual  learning.  

 Vocabulary  factors,  mul%ples,  prime  number,  divide,  mul%ply,  rectangle,  length,  width,  product,  %mes,  lowest  common,  highest  common                        Ac>vity  Process-­‐  Introductory  Ac4vity  Process  The  purpose  of  this  lesson  is  to  introduce  a  revision  finding  a  factor  and  using  mul%ples.  (This  will  demonstrate  what  the  students  know  and  will  determine  how  to  proceed  for  the  lessons  ahead).      Discuss    the  defini%ons  of  mul%ples  and  factors  before  your  students  begin  this  ac%vity.  Make  sure  that  they  understand  that  every  number  is  a  factor  of  itself,  because  if  they  divide  a  number  by  itself,  there  is  no  remainder.  For  example,  12  ÷  12  =  1  without  a  remainder,  so  12  is  a  factor  of  12.  A  prime  number  is  a  number  that  has  only  two  factors,  itself  and  1,  for  example:  5,  7,  13,  and  29.  (Note  that  1  itself  is  not  considered  to  be  a  prime  number.)  Before  the  students  play  the  game,  ask  the  following  ques%ons:  •    Imagine  you  threw  a  4  and  a  6.  Which  squares  could  you  choose  to  cover  with  your  counter?(a  number  with  more  than  two  factors,  a  factor  of  24,  a  mul%ple  of  2,  a  mul%ple  of  3,  a  mul%ple  of  4,  a  mul%ple  of  8,  an  even  number,  or  a  mul%ple  of  6)  •  Imagine  you  need  a  mul%ple  of  5  to  get  four  counters  in  a  row.  Which  throws  of  the  dice  would  give  you  a  mul%ple  

of  5?  (1  and  5,  2  and  5,  3  and  5,  4  and  5,  5  and  5,  6  and    5,  7  and  5,  8  and  5,  or  9  and  5)  

(Adapted  from  h#p://nzmaths.co.nz  /resource/mul4ples-­‐and-­‐factors)  

   

Teacher  background  informa6on  to  be  shared  and  discussed  or  for  students  to  record  into  a  maths  journal/dic6onary:    The  factors  of  a  number  are  any  numbers  that  divide  into  it  exactly.  This  includes  1  and  the  number  itself.  For  example,  the  factors  of  6  are  1,  2,  3  and  6.  The  factors  of  8  are  1,  2,  4  and  8.  For  larger  numbers  it  is  some%mes  easier  to  'pair'  the  factors  by  wri%ng  them  as  mul%plica%ons.  For  example,  24  =  1  x  24  =  2  x  12  =  3  x  8  =  4  x  6  So  the  factors  of  24  are  1,  2,  3,  4,  6,  8,  12  and  24.    Learners    complete  3  factor  and  mul%ple  examples-­‐eg  2  is  a  factor  of  8,  8  is  a  mul%ple  of  2    Ac4vity  Process-­‐Game  On  This  game  could  be  extended  by  asking:  •  What  are  all  the  different  products  you  could  throw  with  the  two  game  dice,  one  labelled  1–6  and  the  other  4–9?  (4,  5,  6,  7,  8,  9,  10,  12,  14,  15,  16,  18,  20,  21,  24,  25,  27,  28,  30,  32,  35,  36,  40,  42,  45,  48,  54)  •  There  are  two  different  ways  of  gegng  a  product  of  12:  throwing  a  3  and  a  4  or  a  2  and  a  6.  Which  other  products  can  you  throw  more  than  one  way  using  the  game  dice?  (8:  1  x  8  or  2  x  4;  16:  2  x  8  or  4  x  4;  18:  2  x        

Page 2:  · ... (length,(width,(product,(%mes ... (24,(amul%ple(of(2,(amul%ple(of(3,(amul%ple(of(4,(amul ... appropriate(units(of(measurementfor(length,(area,(volume

their  own  choosing.  List  the  factors  of  each  number  explored  into  their  maths  books.  (if  you  have  access  to  Signpost  DVD  there  are  some  good  examples  that  can  be  accessed  for  whole  class,  small  group  or  individual  work).  Op6on  2:  this  is  another  lesson  example  which  could  be  used  to  further  explore  factors  and  factor  trees  or  used  in  conjunc%on/ajer  op%on  1.  (there  some  images  on  Google  images  that  can  help  with  those  visual  students  in  the  class)  you  need  to  show  examples    Ac>vity  Process-­‐Using  Factor  Trees.  A  factor  tree  is  a  diagram  used  to  break  down  a  number  by  dividing  it  by  its  factors  un%l  all  the  numbers  lej  are  prime.  A  prime  number  is  a  number  that  is  only  divisible  by  itself  and  the  number  1.  The  number  2  is  the  only  even  prime  number.  The  first  ten  prime  numbers  are  2,  3,  5,  7,  11,  13,  17,  19,  23,  29  …    Problem  -­‐  Cindy  wants  to  find  the  factors  of  36.  She  knows  the  following  %mes  tables  -­‐  36  =  2  x  18  36  =  4  x9  36  =  6  x6.    Draw  a  factor  tree  for  36  =  2  x  18  to  show  the  students  what  the  representa%on  will  look  like.    Then  draw  factor  trees  that  start  with  36  =  4  x  9  and  36  =  6  x6  .  Discuss  why  the  ends  of  the  tree  all  show  36  =  2  x  3  x  3  x  2.  Now  have  the  students  repeat  the  process  by  drawing  factor  trees  using  24  and  explain  that  it  ends  with  2  x  2  x  2  x  3.  Repeat  the  process  with  30      

9  or  3  x  6;  20:  4  x  5  or  5  x  4;  24:  3  x  8  or  4  x  6  or  6  x  4;  30:  5  x  6  or  6  x  5;  36:  4  x  9  or  6  x  6)  •  What’s  the  probability  of  throwing  a  double?  (There  are  36  possible  combina%ons  that  can  be  thrown  with  these  dice,  and  only  3  of  these  are  doubles:  double  4,  5,  or  6.  So  the  probability  of  throwing  a  double  is  or.)  •  Which  squares  in  the  game  are  easier/harder  to  cover?  Can  you  use  the  informa%on  you  have  about  the  possible  products  that  can  be  thrown  to  explain  why?  (Easier  to  cover:  a  number  with  more  than  two  factors  [34  out  of  36  possible  combina%ons  have  more  than  2  factors;  only  5  and  7  don’t],  an  even  number,  a  mul%ple  of  2  [27  out  of  36  possible  combina%ons  are  even  and  are  therefore  also  mul%ples  of  2],  and  a  mul%ple  of  3  [20  out  of  36  possible  combina%ons].  Harder  to  cover:  a  prime  number  [only  2  out  of  36  combina%ons]  and  a  mul%ple  of  7  [only  6  chances  out  of  36].)  (adapted  from  h#p://nzmaths.co.nz/resource/mul4ples-­‐and-­‐factors)    Ac>vity  Process-­‐  Choose  from  the  2  op%ons  to  introduce  the  concept  of  factors  during  the  first  couple  of  days  of  the  week  and  then  proceed  onto  the  op%ons  for  ac%vi%es  involving  mul%ples.    Op6on  1:  this  lesson  will  help  the  students  iden%fy  and  describe  the  factors  of  whole  numbers.  Have  each  student  use  24  cubes  (or  MAB  units)  to  make  a  rectangular  paRern.  Record  the  number  of  cubes  used  on  each  side  of  the  rectangle,  eg:  4  and  6.  Have  students  make  different  rectangles  and  discuss  the  results.  Note  that  1  x  24  =  24,  2  x  12  =  24,  3  x  8  =  24,  Ask  students  to  use  the  cubes  to  explore  numbers  of    

 

Ac>vity  Process-­‐Introducing  Mul6ples.  Op6on  1  Give  prac%ce  at  coun%ng  by  twos,  fives,  tens  and  fours.  As  we  count  by  fours,  we  say  (use  the  language)    mul%ples  of  fours.  Make  mul%ples  using  rows  of  place-­‐value  (MAB)  ones.  Explain  that,  if  4  is  a  factor  of  28,  then  28  is  a  mul%ple  of  4.  Discuss  now  with  the  students  that  a  mul%ple  is  the  product  of  any  two  or  more  of  its  factors.  Explain  and  demonstrate  on  the  board  that  some  numbers  are  mul%ples  of  several  other  numbers.  For  example  -­‐  24  is  a  mul%ple  of  3,  4,  6,  12  and  24.  An  extension  ac%vity  to  this  is  located  on  the  Signpost  Maths  DVD  5.  Extension  -­‐  use  a  calculator  to  list  the  first  ten  mul%ples  of  14  and  21.  Encourage  the  students  to  use  the  constant  mul%plier.  i.e:  14  X  X  1  =  2  =  etc.  answer:  The  first  three  common  mul%ples  of  14  and  21  are  42,  84  and  126.    Discuss  and  review  the  term  ‘mul%ple.’  24  is  a  mul%ple  of  8  because  3  x  8  =  24.  Have  the  students  present  some  other  example  they  have  discovered  on  the  IWB    

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Op6on  2:    Review  the  terms  ‘factors’  and  ‘mul%ples’  and  elicit  from  the  students  what  knowledge  they  have  to  these  two  terms.  Explain  that  the  lesson  will  be  focussing  on  mul%ples  of  whole  numbers  and  give  some  examples  using  some  mul%ples  of  5  -­‐  5,  10,  15,  25,  50,  etc  to  start  the  students  thinking  paRern.  Now  move  on  and  draw  on  the  board  a  Concept  box  (arrays)  -­‐  explain  that  this  is  another  way  of  represen%ng  and  loca%ng  factors  and  mul%ples  of  numbers.  Use  the  example  of  3  x  8  =  24  so  three  rows  of  eight  =  24  and  draw  the  box  to  show  the  paRern  (especially  for  visual  learners)  and  keep  repea%ng  a  line  of  eight  to  introduce  the  next  mul%ple.  You  might  like  to  start  with  a  smaller  number  depending  on  whiteboard  size  or  alterna%vely  give  the  students  some  grid  paper  and  allow  them  to  copy  and  keep  mul%plying.  This  is  beneficial  for  differen%ated  learners.  For  these  differen%ated  learners  you  may  like  to  give  them  a  copy  of  a  mul%plica%on  table  to  help  them  complete  the    work.  Prac%ce  with  lots  of  examples  (consolida%on)  by  breaking  students  up  into  small  groups  and  have  them  write  onto  large  s%cky  notes  (poster  size)  all  the  mul%ples  for  a  given  number  up  to  100/150.  For  example  -­‐  one  group  might  be  doing  mul%ples  of  3’s,  another  mul%ples  of  8,  etc.  Spend  some  %me  at  the  end  of  the  lesson  sharing  these  and  display  as  living  charts  around  the  classroom.    Op6on  3:    Extend  the  previous  ac%vi%es  by  introducing  the  terms  of  lowest  and  highest  common  factors  and  mul%ples.  Revise  with  students  what  is  a  factor  and  what  is  a  mul%ple.  Place  some  examples  on  the  board  and  look  at  what  is  the  lowest  common  factor  -­‐  explain  that  all  whole  numbers  have  1  and  itself  as  factors  but  you  want  to  find  the  lowest  common  factor  other  than  1.  For  example  -­‐  12  and  15  the  lowest  common  factor  would  be  3.  Examples  of  board  work  below.  Also  instruct  and  look  at  finding  the  lowest  common  mul%ple  of  two  numbers  eg:  12  and  24  would  be  48  and  the  highest  common  mul%ple  between  100  and  200.  

Extensions  and  Varia>ons  Ac>vi>es    •  Ask  students  use  a  calculator  to  find  the  product  of  any  two  numbers  less  than  20.  They  can  ask  a  partner  to  use  the  calculator  to  find  the  factors  of  a  product?  

This  could  also  be  repeated  for  mul%ples.  

•  Ask  students  solve  problems  using  whole  numbers.  They  will  need  to  iden%fy  and  describe  the  factors  and  mul%ples  of  whole  numbers  to  solve  these  problems.  

•  Ask  your  high  level  learners  make  A4  size  (living  charts  if  needed)  charts  dealing  with  mul%ples  above  150  etc.  This  can  also  be  done  for  factors.  

•  Have  students  work  with  calculators  to  mul%ply  larger  numbers  eg:  24  x  25  using  factors  and  mul%ples  -­‐  24  is  4  x  6  x  25  or  2  x  12  x  25,  or  (2  x  3)  x  (  4  x  25  )  etc.  

•  Factor  crossword    

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Assessment-­‐  By  the  end  of  Year  5,  students  solve  simple  problems  involving  the  four  opera%ons  using  a  range  of  strategies.  They  check  the  reasonableness  of  answers  using  es%ma%on  and  rounding.  Students  iden%fy  and  describe  factors  and  mul%ples.  They  explain  plans  for  simple  budgets.  Students  connect  three-­‐dimensional  objects  with  their  two-­‐dimensional  representa%ons.  They  describe  transforma%ons  of  two-­‐dimensional  shapes  and  iden%fy  line  and  rota%onal  symmetry.  Students  compare  and  interpret  different  data  sets.  Students  order  decimals  and  unit  frac%ons  and  locate  them  on  number  lines.  They  add  and  subtract  frac%ons  with  the  same  denominator.  Students  con%nue  paRerns  by  adding  and  subtrac%ng  frac%ons  and  decimals.  They  find  unknown  quan%%es  in  number  sentences.  They  use  appropriate  units  of  measurement  for  length,  area,  volume,  capacity  and  mass,  and  calculate  perimeter  and  area  of  rectangles.  They  convert  between  12  and  24  hour  %me.  Students  use  a  grid  reference  system  to  locate  landmarks.  They  measure  and  construct  different  angles.  Students  list  outcomes  of  chance  experiments  with  equally  likely  outcomes  and  assign  probabili%es  between  0  and  1.  Students  pose  ques%ons  to  gather  data,  and  construct  data  displays  appropriate  for  the  data.      Independent  Assessment-­‐Students  choose  one  op6on  to  complete    Op%on  1  A  furniture  shop  sells  chairs  in  sets  of  four.  Which  of  the  following  numbers  of  chairs  would  make  complete  sets.  10,  12,  15,  20,  22,  28,  35?  Jus%fy    

 

 

Complete  a  Factor  Crossword    Across  1.  Mul%ples  of  7  3.  Mul%ples  of  5  5.  Factor  of  24  6.  Has  factors  of  2  and  5  9.  Factor  of  42    Down  1.  Number  %mes  10  divides  by  5  2.  Mul%ple  of  10  4.  Factor  of  27  6.  Factor  of  27  8.  Mul%ple  of  2    Digital  Learning  –    The  following  digital  resources  can  be  used  for  reinforcement  or  consolida%on  ac%vi%es  at  teachers  discre%on  Promethean  Planet  has  a  number  of  various  flipcharts  available  for  the  IWB  and  below  are  some  screenshots  with  the  names  of  some  of    the  flipcharts  available.  

Inves>ga>on:    You  have  been  asked  to  create  a  display  of  field  trip    photos  for  a  class  webpage.  Create  a  diagram  of  your  various  layout  op%ons  for  arranging  them  on  the  screen.      Select  one  and  jus%fy  why  it  would  best  fit  suit  the  screen?  

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your  answer  using  a  green  fish  example    Op%on  2  A  gardener  has  12  flowering  plants  that  he  wants  to  plant  in  equal  rows  at  the  front  entrance  to  the  garden.  What  are  all  the  ways  he  can  arrange  the  flowers  in  equal  rows?  Jus%fy  your  answer  using  a  green  fish  example    Op%on  3  A  concert  organiser  has  200  seats  he  needs  to  arrange  in  rows  with  access  aisles  on  the  sides  and  one  in  the  middle.  What  are  his  possible  op%ons  for  moving  concert  aRendees  quickly  and  safely  in  and  out  of  their  seats.  Jus%fy  your  choice  of  most  efficient  sea%ng  arrangement  you  would  recommend  he  use.  Jus%fy  your  answer  using  a  green  fish  example                        

 

 

 

 

Background-­‐Learners  need  to  be  able  to  break  down  a  number  into  all  its  factors.  

Students  need  to  be  able  to  solve  number  problems  that  they  will  come  across  in  everyday  life.  In  order  to  achieve  this  they  need  to  have  a  knowledge  bank  of  number  facts  which  will  allow  them  to  develop  effec%ve  methods  of  calcula%on.  To  do  this  they  need  to  create,  use  and  explore  number  paRerns  in  a  wide  range  of  ac%vi%es  which  include  prac%cal,  mental  and  formal  work  in  different  contexts.  For  example  purchasing  mul%ple  items  when  shopping  or  planning  a  celebra%on  etc.      

 

 

 

Was    the

 learne

r  able  to   Above   Expected   Below  

Iden%fy  appropriate  mathema%cal  concepts,  skills  and  strategies  necessary  to  prove  a  solu%on  

Use  skills  and  strategies  that  lead  to  a  simple  solu%on  

Use  strategies  that  lead  to  an  unsuccessful  solu%on    

Demonstrate  the  difference  between  factors  and  mul%ples  

Use  simple  factors  and  mul%ples   Unsucessfully  use  factors  and  mul%ples  

Able  to  use  calculator  func%ons  effec%vely  to  check  answers    

Able  to  use  calculator  independently  to  check  answers  

Unable  to  use  calculator  independently  to  check  answers