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PROGRAMMABLE LOGIC CONTROLLERS 7ELEMENT MATHEMATICS & SCIENCE LESSON MANUFACTURING TECHNOLOGY AND ROBOTICS Jessica L. Campbell Ben Franklin Career and Technical Center Dunbar, WV

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Microsoft Word - PLCLesson.docx  7 -­ E L EMEN T  MATHEMAT I C S   &
S C I E N C E   L E S S ON
MANUFACTURING  TECHNOLOGY  AND   ROBOTICS

SUMMARY  ...........................................................................................................................................  3
ELEMENT  ONE:    IDENTIFY  LEARNING  GOALS  .........................................................................................  4   CAREER  AND  TECHNICAL  STANDARDS:  ..............................................................................................................  4   21ST  CENTURY  SKILLS:  ...................................................................................................................................  4   ENGLISH  LANGUAGE  ARTS  STANDARDS  ............................................................................................................  4   MATHEMATICS  STANDARDS  ...........................................................................................................................  5   SCIENCE  STANDARDS  .....................................................................................................................................  5
ELEMENT  TWO:  PRE-­ASSESSMENT  ........................................................................................................  6   ELEMENT  TWO:  PRE-­ASSESSMENT  RUBRIC  .......................................................................................................  8
ELEMENT  THREE:    EMBEDDED  CONTEXTUAL  EXAMPLES  .....................................................................  11   ELEMENT  THREE:  EMBEDDED  CONTEXTUAL  EXAMPLES  -­  SOLUTIONS  ...................................................................  13
ELEMENT  FOUR:  RELATED  CONTEXTUAL  EXAMPLES  ...........................................................................  15   ELEMENT  FOUR:  RELATED  CONTEXTUAL  EXAMPLES  -­  SOLUTIONS  ........................................................................  17
ELEMENT  FIVE:    TRADITIONAL  EXAMPLES  ...........................................................................................  20   ELEMENT  FIVE:    TRADITIONAL  EXAMPLES-­  SOLUTIONS  ......................................................................................  22
ELEMENT  SIX:  DEMONSTRATE  UNDERSTANDING  ................................................................................  24

Summary   This  cross-­curricular  lesson  uses  project-­based  learning  to  meet  mathematics  and  science   standards  with  applications  to  Programmable  Logic  Controllers  as  used  in  industrial  processes.     The  lesson  follows  the  seven  element  format.         Element  One  –  Identify  Learning  Goals   Refer  to  the  standards  listed  at  the  beginning  of  the  lesson.    Throughout  the  project,  the   following  questions  guide  student  learning:
•   What  are  programmable  logic  controllers  and  how  are  they  used  in  industrial   applications?
•   How  can  I  define  and  evaluate  logical  statements?   •   How  can  I  use  mathematical  modeling  to  solve  real  world  problems?   •   What  scientific  principles  can  be  applied  to  guide  design  and  reach  solutions?
Element  Two  –  Pre-­Assess  Learning   Students  individually  complete  the  Pre-­Assessment,  which  serves  as  a  formative  assessment  to   identify  common  obstacles,  misconceptions,  and  gaps  in  learning.    The  instructor  may  use  the   pre-­assessment  rubric  to  offer  feedback  aimed  to  elevate  student  understanding.    Group   students  based  on  pre-­assessment  responses  for  team  activities.         Element  Three  –  Embedded  Contextual  Examples   Students  model  real-­world  problems  in  teams  assigned  homogeneously  based  on  pre-­ assessment  responses.    Student  teams  complete  a  “gallery  walk”  to  observe  and  offer  feedback   to  each  other.    Instructor  closes  with  plenary  discussion.     Element  Four  –  Related  Contextual  Examples   Students  de-­contextualize  then  re-­contextualize  to  analyze  and  solve  problem  scenarios  related   to  the  original  problem.       Element  Five  –  Traditional  Examples   Students  encounter  and  solve  problems  as  they  would  classically  appear  on  standardized  tests.       Element  Six  –  Students  Demonstrate  Understanding   Students  demonstrate  understanding  of  key  ideas  through  hands-­on  project  based  learning   activities.       Element  Seven  –  Formal  Post-­Assessment   Assessment  items  requiring  mastery  of  standards  addressed  through  project  are  completed  by   each  student  individually.
Element  One:    Identify  Learning  Goals   Career  and  Technical  Standards:   MA  1870  Industrial  Equipment  Maintenance   1871.32  Explain  the  purpose  of  Programmable  Logic  Controllers  (PLC)  and  their  applications  in   industrial  locations.   1871.33  Identify  the  component  parts  of  a  PLC  system.   1871.34  Explain  how  to  program  and  trouble  shoot  PLC’s.     MA  1880  Industrial  Technology   1808.1    Students  will  demonstrate  a  knowledge  of  the  purpose,  function,  and  application  of   programmable  logic  controllers  (PLCs)  &  electronic  controls.   1808.7    Students  will  define  a  logical  truth  table.   1808.9    Students  will  evaluate  logical  statements.     21st  Century  Skills:   21C.O.5-­8.1.LS.2  Student  interprets  abstract  visuals  and  creates  products  (e.g.  digital   storytelling)  that  reflect  a  growing  understanding  of  visual  language  and  require  the  effective   use  of  tools  (e.g.  cropped  photos,  original  charts  and  graphs,  well-­chosen  images  from   databases,  video  clips).     21C.O.5-­8.2.LS.1  Student  engages  in  a  critical  thinking  process  that  supports  synthesis  and   conducts  evaluations  by  applying  comprehensive  criteria.     21C.O.5-­8.3.LS.2  Student  is  flexible  in  approach  to  solving  problems  and  completing  tasks,   considers  alternative  methods,  solutions  and  perspectives,  abandons  strategies  that  do  not   work,  and  reallocates  time  and  resources  as  priorities  change.     21C.O.5-­8.3.LS.6  Student  maintains  focus  on  larger  project  goal,  frames  appropriate  questions,   reflects  on  possible  courses  of  action  and  their  likely  consequences,  develops  and  initiates  a   plan  of  action  with  appropriate  smaller  objectives  and  benchmarks,  and  submits  the  completed   project  when  due.     English  Language  Arts  Standards     ELA.6.10  Determine  the  meaning  of  words  and  phrases  as  they  are  used  in  an  informational   text,  including  figurative,  connotative,  and  technical  meanings.     ELA.6.15  Integrate  information  presented  in  different  media  or  formats  (e.g.,  visually  and/or   quantitatively)  and  in  words  to  develop  a  coherent  understanding  of  a  topic  or  issue.     ELA.6.31  Interpret  information  presented  in  diverse  media  and  formats  (e.g.,  visually,   quantitatively,  and/or  orally)  and  explain  how  it  contributes  to  a  topic,  text,  or  issue  under   study.
Mathematics  Standards   7.EE.B.4  Use  variables  to  represent  quantities  in  a  real-­world  mathematical  problem,  and   construct  simple  equations  and  inequalities  to  solve  problems  by  reasoning  about  the   quantities.     BF.A.1.A    Determine  an  explicit  expression,  recursive  process,  or  steps  for  calculation  from  a   context.         LE.B.5  Interpret  parameters  in  a  linear  or  exponential  function  in  terms  of  a  context.       Science  Standards   S.6-­8.L.4  Students  will  determine  the  meaning  of  symbols,  key  terms,  and  other  domain-­specific   words  and  phrases  as  they  are  used  in  a  specific  scientific  or  technical  context  relevant  to   grades  6–8  texts  and  topics.         S.6-­8.L.7  Integrate  quantitative  or  technical  information  expressed  in  words  in  a  text  with  a   version  of  that  information  expressed  visually  (e.g.,  in  a  flowchart,  diagram,  model,  graph,  or   table).         S.6-­8.ETS.1  Define  the  criteria  and  constraints  of  a  design  problem  with  sufficient  precision  to   ensure  a  successful  solution,  taking  into  account  relevant  scientific  principles  and  potential   impacts  on  people  and  the  natural  environment  that  may  limit  possible  solutions.

3.   Create  a  truth  table  for  each  statement  below:     A.   Bi-­conditional  statement:    x  =  12  iff  x  *  0.25  =  3   B.   Converse:    x  *  0.25  =  3  iff  x  =  12   C.   Inverse:    x  ≠  12  iff  x  *  0.25  ≠  3   D.   Contrapositive:    x  *  0.25  ≠  3  iff  x  ≠  12

5.   Complete  the  logic  table  for  p  XOR  q  (pq)
p   q   (pq)
Misconceptions   Recommended   Feedback
1.   Describe  the   scenario  depicted   by  the  logic   ladder.
If  switch  1  is  on  and   switch  2  is  off,  the   irrigation  system  will   turn  on.    If  switch  2  is   on  and  switch  1  is   off,  the  irrigation   system  will  turn  on.
Students  are  unsure   about  symbolic   representation.     Students  are   confused  about  logic   of  process.
How  do  symbols   represent  actions?   Which  symbols  can   be  used  to  represent   “off”  or  “on?”   What  variables  and   symbols  are   encountered  when   you  trace  from  input   to  output?
2.   Explain  the   process  the  flow   chart  illustrates.
The  programming   interface  is  used  to   program  conditions   for  automated   actions.    The  CPU   receives  the  program   and  exchanges   information  with  the   I/O  section.    Sensor   input  is  received  by   the  I/O  section.    The   I/O  section  outputs   action  commands   based  on  the   program  and   information  received   from  the  sensors.
Student   misunderstands   direction  of  flow   chart  arrows.     Student   misunderstands   function  of  one  or   more  components.
Which  arrows   represent  data  being   sent?    Which   represent  data  being   received?    Which   components  receive   data,  and  from   where?    Which   components  send   data,  and  where  do   they  send  it  to?     What  context  clues   or  known  vocabulary   can  be  used  to  infer   the  function  of  each   component?
3.   Create  a  truth   table  for  each   statement  below:
A.   Bi-­conditional   statement:    x  =  12   iff  x  *  0.25  =  3
B.   Converse:    x  *   0.25  =  3  iff  x  =  12
C.   Inverse:    x  ≠  12   iff  x  *  0.25  ≠  3
A,  B,  C,  D.       p   q   p

Student  is  unsure   about  notation.     Student  is  unsure   how  to  construct  a   truth  table.     Student  incorrectly   evaluates  truth   values.
What  does  the   notation  used  in  the   problem  mean?     What  are  the  column   headings  of  a  truth   table?    How  many   possibilities  are  there   for  the  truth  values   of  p  and  q?     Write  the  statement   for  each  truth  value   of  p  and  q.    When
D.   Contrapositive:    x   *  0.25  ≠  3  iff  x  ≠   12
you  read  the   statements  as   complete  sentences,   how  can  you   determine  whether   each  is  true  or  false?

1.   Both  wheel   motors  are  on,   therefore  (2)  the   robot  moves   forward  for  2.5   seconds  (twenty-­ five  one-­tenth   second   increments).
3.   Both  wheel   motors  turn  off,   then  (4)  only  the   left  wheel  motor   turns  on  for  half  a   second.
4.   The  left  wheel   motor  turns  on   causing  the  robot   to  turn  right
5.   The  robot  turns   right  for  0.5   seconds  (five   one-­tenth  second   increments).
6.   The  left  wheel   motor  turns  off
7.   Both  wheel   motors  turn  on
8.   The  robot  moves   forward  for  one   second.
9.   Both  wheel   motors  turn  off.
Student  is  unsure   about  what  action   occurs  when  one  or   both  wheel  motors   are  on.     Student  is  confused   about  time   increments
What  happens  when   both  wheel  motors   are  on?       What  happens  when   the  left  wheel  spins   but  the  right  wheel   remains  stationary?     What  happens  when   the  right  wheel  spins   but  the  left  wheel   remains  stationary?     What  is  25  *  0.1?   How  many  one-­tenth   second  increments   are  in  one  second?
5.   Complete  the   logic  table  for  p   XOR  q  (pq)   p   q   (p
q)
p   q   (p q)

Student  is  unsure   about  meaning  of   notation   Student  is  certain   about  meaning  of   logical  statements

Element  Three:    Embedded  Contextual  Examples     Pair  each  card  from  set  A  to  its  best  match  from  set  B.    Explain  your  reasoning  and  justify  your   selection.
Card  Set  A   Card  Set  B
If  (x,y)  =  (0,90)  then  turn  90o

If  <Switch  1  AND  NOT  Switch  2>  input,   OR  <Switch  2  AND  NOT  Switch  1>  input
then  output  irrigation.

If  the  engine  temperature  is  too  hot,   then  the  temperature  gauge  indicates
red.   If  the  engine  temperature  is  too  hot,   then  the  temperature  gauge  does  not
indicate  red.   If  the  engine  temperature  is  not  too
hot,  then  it  is  possible  the  temperature   gauge  may  indicate  red.
If  the  engine  temperature  is  not  too   hot,  then  the  temperature  gauge  does
not  indicate  red.

If  x  =  8  then  x  *  0.75  =  6   If  x  *  0.75  =  6  then  x  =  8   If  x  ≠  8  then  x  *  0.75  ≠  6   If  x  *  0.75  ≠  6  then  x  ≠  8
Conditional  Statement
Converse   If  q  then  p   Inverse
If  not  p  (~p)  then  not  q  (~q)     Contrapositive   If  ~q  then  ~p

Element  Three:  Embedded  Contextual  Examples  -­  Solutions   Card  Set  A   Card  Set  B
If  (x,y)  =  (0,90)  then  turn  90o   Conditional  Statement

If  the  engine  temperature  is  too  hot,  then   the  temperature  gauge  indicates  red.
If  the  engine  temperature  is  too  hot,  then   the  temperature  gauge  does  not  indicate
red.   If  the  engine  temperature  is  not  too  hot,   then  it  is  possible  the  temperature  gauge
may  indicate  red.   If  the  engine  temperature  is  not  too  hot,   then  the  temperature  gauge  does  not
indicate  red.

If  x  =  8  then  x  *  0.75  =  6   If  x  *  0.75  =  6  then  x  =  8   If  x  ≠  8  then  x  *  0.75  ≠  6   If  x  *  0.75  ≠  6  then  x  ≠  8
Conditional  Statement     If  hypothesis  (p)  then  conclusion  (q)
Converse   If  q  then  p   Inverse
If  not  p  (~p)  then  not  q  (~q)     Contrapositive   If  ~q  then  ~p
Programmable  Logic  Controller

Element  Four:  Related  Contextual  Examples   1.   Which  script  would  program  a  robot  to  carry  out  the  instructions  below?  Let  Aux1  represent
the  left  wheel  motor  and  Aux2  represent  the  right  wheel  motor  of  the  robot.    Explain  your   reasoning  in  complete  sentences.
Instructions:    The  robot  should  move  forward  for  5  seconds,  then  make  a  right-­hand  (clockwise)   turn  for  4.5s.    Next,  the  robot  should  move  forward  for  10s  then  perform  a  left-­hand   (counterclockwise)  turn  for  4.5s.    It  should  finally  drive  forward  5s.
A.         B.         C.       D.                                         2.   Write  the  converse,  inverse,  and  contrapositive  of  the  conditional  statement
If  the  soil  moisture  has  dropped  below  the  lower  control  limit  then  the  irrigation  system   turns  on.
3.   Use  logic  symbols  from  the  table  below  along  with  the  variables  p  and  q  to  represent  each   statement.
Logic   Connective   Operator  Symbol   Conditional   if  then   à   Biconditional   if  and  only  if   ⇔   Negation   NOT   ~   Conjunction   AND   Λ   Disjunction   OR  (either  p,  q,  or  both)   V   Exclusive  or   either…or  but  not  both  XOR   ⊕
A.   If  I  pay  attention  (p),  then  I  achieve  more  (q).   B.   The  robot  turns  (p)  if  and  only  if  exactly  one  wheel  is  moving  (q).     C.   The  switch  is  open  (~p)  and  the  light  is  off  (~q).     D.   Either  the  switch  is  closed  or  the  button  is  depressed.   E.   Either  the  material  is  too  hot  or  the  material  is  too  cold.
4.   Complete  the  logic  table  for  p  AND  q  (p∧q)
p   q   (p∧q)
5.   Interpret  the  logic  ladder  below  and  describe  the  process  in  complete  sentences  using  logic
notation.
Element  Four:  Related  Contextual  Examples  -­  Solutions   1.   Which  script  would  program  a  robot  to  carry  out  the  instructions  below?  Let  Aux1
represent  the  left  wheel  and  Aux2  represent  the  right  wheel  of  the  robot.    Explain  your   reasoning  in  complete  sentences.
Instructions:    The  robot  should  move  forward  for  5  seconds,  then  make  a  right-­hand  (clockwise)   turn  for  4.5s.    Next,  the  robot  should  move  forward  for  10s  then  perform  a  left-­hand   (counterclockwise)  turn  for  4.5s.    It  should  finally  drive  forward  5s.
B.         B.         C.       D.                                     B  is  the  correct  script  for  the  instruction  set  given.    Both  wheel  motors  turn  on  for  fifty  one-­ tenth  second  increments,  or  5  seconds.    While  both  wheels  are  turning,  the  robot  is  moving   forward.    Both  wheels  stop  then  the  left  wheel  turns  for  4.5  seconds.    While  only  the  left  wheel   is  spinning,  the  robot  is  turning  right  (clockwise).    The  left  wheel  motor  turns  off  then  both   wheel  motors  turn  on  for  ten  seconds,  meaning  the  robot  moves  forward  for  ten  seconds.     Next,  both  wheels  turn  off  and  only  the  right  wheel  spins  for  4.5  seconds.    While  only  the  right   wheel  is  spinning,  the  robot  is  turning  left  (counterclockwise).    The  right  wheel  turns  off  then   both  wheel  motors  turn  on  causing  the  robot  to  move  forward  for  5  seconds.         The  script  given  in  A  produces  mirrored  results;  for  example,  the  robot  would  be  turning   counterclockwise  when  it  should  be  turning  clockwise.    Neither  script  C  nor  D  would  cause  the   robot  to  carry  out  the  correct  instruction  set.
2.   Write  the  converse,  inverse,  and  contrapositive  of  the  conditional  statement   If  the  soil  moisture  has  dropped  below  the  lower  control  limit  then  the  irrigation  system   turns  on.     Converse:  If  the  irrigation  system  turns  on  then  the  soil  moisture  has  dropped  below  the   lower  control  limit.
Inverse:    If  the  soil  moisture  has  not  dropped  below  the  lower  control  limit  then  the   irrigation  system  does  not  turn  on.     Contrapositive:    If  the  irrigation  system  does  not  turn  on  then  the  soil  moisture  has  not   dropped  below  the  lower  control  limit.
3.   Use  logic  symbols  from  the  table  below  along  with  the  variables  p  and  q  to  represent   each  statement.
Logic   Connective   Operator  Symbol   Conditional   if  then   à   Biconditional   if  and  only  if   ⇔   Negation   NOT   ~   Conjunction   AND   Λ   Disjunction   OR  (either  p,  q,  or  both)   V   Exclusive  or   either…or  but  not  both   ⊕
A.   If  I  pay  attention  (p)  then  I  achieve  more  (q).   B.   The  robot  turns  (p)  if  and  only  if  exactly  one  wheel  is  moving  (q).     C.   The  switch  is  open  (~p)  and  the  light  is  off  (~q).     D.   Either  the  switch  is  closed  or  the  button  is  depressed.   E.   Either  the  material  is  too  hot  or  the  material  is  too  cold.     A.   Paying  attention  à  higher  achievement.    p  à  q   B.   Robot  turns  ⇔  exactly  one  wheel  moving.    p  ⇔  q.   C.   Open  switch  ∧  light  off.    ~p    ~q.     D.   The  switch  is  closed,  the  button  is  depressed,  or  both.    p  ∨  q.     E.   The  material  is  too  hot  or  the  material  is  too  cold,  but  it  is  not  both  too  hot  and  too  cold.
p  ⊕  q.
4.   Complete  the  logic  table  for  p  AND  q  (pq)
p   q   (p∧q)   T   T   T   T   F   F   F   T   F   F   F   F
5.   Interpret  the  logic  ladder  below  and  describe  the  process  in  complete  sentences  using
logic  notation.

1.   Write  the  given  statement  as  a  conditional  statement,  then  write  its  converse,  inverse,   and  contrapositive.       Strong  effort  (p)  is  directly  related  to  high  achievement  (q).
2.   Create  a  flow  chart  for  the  geometric  proof  shown  below:
Statements   Reasons   ABCD  is  a  rectangle   Given   Line  segment  AB  is  congruent  to  line  segment   BD.      ≅
Definition  of  a  rectangle
Line  segment  AD  is  congruent  to  line   segment  BC.      ≅
Definition  of    a  rectangle
∠ = ∠ = ∠ = ∠ = 90°   Definition  of  a  rectangle   ∠   ≅  ∠   ≅  ∠   ≅  ∠   Definition  of  congruence   ≅     Side  –  Angle  –  Side  (SAS)  postulate
3.   Create  a  truth  table  for  the  statement:  An  angle  is  acute  if  and  only  if  (iff)  the  angle   measure  is  less  than  90°.
4.   Write  four  of  your  own  conditional  logic  statements.

5.   Create  a  truth  table  for  one  of  your  conditional  statements.
Element  Five:    Traditional  Examples-­  Solutions   1.   Write  the  given  statement  as  a  conditional  statement,  then  write  its  converse,  inverse,
and  contrapositive.       Strong  effort  (p)  is  directly  related  to  high  achievement  (q).     Answers  may  vary.    For  example,     Conditional  statement:    If  strong  effort  is  applied,  then  high  achievement  will  be   accomplished.       Converse:    If  high  achievement  is  accomplished,  then  strong  effort  has  been  applied.     Inverse:    If  strong  effort  is  not  applied,  then  high  achievement  will  not  be  accomplished.     Contrapositive:    If  high  achievement  is  not  accomplished,  then  strong  effort  has  not   been  applied.
2.   Create  a  flow  chart  for  the  geometric  proof  shown  below:
Statements   Reasons   ABCD  is  a  rectangle   Given   Line  segment  AB  is  congruent  to  line  segment   BD.      ≅
Definition  of  a  rectangle
Line  segment  AD  is  congruent  to  line   segment  BC.      ≅
Definition  of    a  rectangle

3.   Create  a  truth  table  for  the  statement:  An  angle  is  acute  if  and  only  if  (iff)  the  angle
measure  is  less  than  90°.   p   q   p    q   T   T   T   T   F   F   F   T   F   F   F   T
4.   Write  four  of  your  own  conditional  logic  statements.
Answers  will  vary.    A  conditional  statement  has  the  form  if  (condition)  then  (conclusion).   Examples:    If  I  oversleep  then  I  miss  the  bus.    If  I  forget  my  homework,  then  I  receive  a   zero.    If  I  apply  my  best  effort,  then  I  will  perform  better.    If  I  take  advantage  of   opportunities  to  do  well,  then  I  will  have  more  opportunities.    If  I  meet  challenges  with  a   positive  attitude,  then  I  have  a  higher  chance  of  persevering.
5.   Create  a  truth  table  for  one  of  your  conditional  statements.     Answers  will  vary.    For  example,     If  I  oversleep,  then  I  will  miss  the  bus.

Element  Seven:    Post-­Assessment
1.   Compose  a  script  that  will  cause  a  two-­wheeled  robot  to  move  forward  for  ten  seconds,   turn  right  for  five  seconds,  move  forward  again  for  fifteen  seconds,  then  turn  left  for  2.5   seconds.    Finally,  the  robot  should  stop.    Designate  the  left  wheel  motor  as  AUX  1  and   the  right  wheel  motor  as  AUX  2.    Compute  time  in  one-­tenth  increments  of  a  second.

3.   Complete  the  truth  table  for  p  OR  q  (p  ∨  q).
p   q   p  ∨  q
4.   Write  a  conditional  statement  based  on  the  statement  given,  then  write  its  converse,
inverse,  and  contrapositive.     Meeting  challenges  with  a  positive  attitude  increases  the  chance  of  perseverance.

Element  Seven:    Post-­Assessment  –  Answer  Key   1.   Compose  a  script  that  will  cause  a  two-­wheeled  robot  to  move  forward  for  ten  seconds,
turn  right  for  five  seconds,  move  forward  again  for  fifteen  seconds,  then  turn  left  for  2.5   seconds.    Finally,  the  robot  should  stop.    Designate  the  left  wheel  motor  as  AUX  1  and   the  right  wheel  motor  as  AUX  2.    Compute  time  in  one-­tenth  increments  of  a  second.       1.   AUX  1,  2  ON   2.   Wait  100   3.   AUX  1,  2  OFF   4.   AUX  1  ON   5.   WAIT  50   6.   AUX  1  OFF   7.   AUX  1,  2  ON   8.   WAIT  150   9.   AUX  2  ON   10.  WAIT  25   11.  AUX  2  OFF
2.   Explain  the  flow  chart  given  below  in  a  paragraph  summary.
It  is  given  that  ABCD  is  a  rectangle.    By  the  definition  of  a  rectangle,  opposite  sides  are   congruent  and  the  interior  angles  are  right  angles.    All  right  angles  are  congruent.    It  is  proven   that  interior  triangles  created  by  the  diagonals  are  congruent  by  the  Side-­Angle-­Side  Postulate.
3.   Complete  the  truth  table  for  p  OR  q  (p  ∨  q).
p   q   p  ∨  q   T   T   T   T   F   T   F   T   T   F   F   F
4.   Write  a  conditional  statement  based  on  the  statement  given,  then  write  its  converse,
inverse,  and  contrapositive.     Meeting  challenges  with  a  positive  attitude  increases  the  chance  of  perseverance.     Conditional  statement:    If  challenges  are  met  with  a  positive  attitude,  then  the  chance   of  perseverance  is  increased.     Converse:    If  the  chance  of  perseverance  is  increased,  challenges  are  met  with  a  positive   attitude.   Inverse:    If  challenges  are  not  met  with  a  positive  attitude,  then  the  chance  of   perseverance  is  not  increased.       Contrapositive:    If  the  chance  of  perseverance  is  not  increased,  then  challenges  are  not   met  with  a  positive  attitude.

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