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Problem solving and metacognition 1 Copyright © 2016 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, Agile Mind, Inc. PROBLEM SOLVING AND METACOGNITION Lesson 1: Thinking about thinking LESSON 1: OPENER Consider some of the problems you have tackled in this course. The Banquet Table Problem The Van Rental Problem The Pond Border Problem 1. How did you feel when you first started working on each problem? Were you confident? Confused? Worried? 2. How did you feel after you solved each problem? LESSON 1: CORE ACTIVITY The Frozen Yogurt Store Problem You and your friend go to a frozen yogurt store. You both like to get frozen yogurt cones with different toppings. The store has a sign showing the different kinds of cones, yogurt, and toppings you can buy. You and your friend wonder how many different onetopping cones you can make. Work with your partner to solve the following problems. 1. Find all the different combinations of cones, yogurt, and toppings you can make and explain how you know you have found all of them.

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TRANSCRIPT Problem  solving  and  metacognition   1

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

PROBLEM SOLVING AND METACOGNITION Lesson 1: Thinking about thinking

LESSON 1: OPENER Consider  some  of  the  problems  you  have  tackled  in  this  course.

• The  Banquet  Table  Problem

• The  Van  Rental  Problem

• The  Pond  Border  Problem

1. How  did  you  feel  when  you  first  started  working  on  each  problem?  Were  you  confident?  Confused?  Worried?

2. How  did  you  feel  after  you  solved  each  problem?

LESSON 1: CORE ACTIVITY

The  Frozen  Yogurt  Store  Problem

You  and  your  friend  go  to  a  frozen  yogurt  store.  You  both  like  to  get  frozen  yogurt  cones  with  different  toppings.  The  store  has  a  sign  showing  the  different  kinds  of  cones,  yogurt,  and  toppings  you  can  buy.  You  and  your  friend  wonder  how  many  different  one-­‐topping  cones  you  can  make.

Work  with  your  partner  to  solve  the  following  problems.

1. Find  all  the  different  combinations  of  cones,  yogurt,  and  toppings  you  can  make  and  explain  how  you  know  you  have    found  all  of  them. 2   Problem  solving  and  metacognition

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

2. How  would  your  numbers  change  if  the  store  added  a  waffle  cone?  Explain.

Now  that  you  have  completed  the  Frozen  Yogurt  Store  Problem,  talk  with  your  partner  about  what  happened  while  you  were  solving  the  problem  and  answer  the  following  questions.

a. Did  you  feel  confused?  Did  you  know  exactly  how  to  get  started?  Think  of  a  few  words  to  describe  your  feelings.

b. Did  you  make  a  plan  or  just  start  making  different  combinations  of  cones,  yogurt,  and  toppings?  If  you  came  up  with  a  plan,  describe  it.

c. Did  the  approach  you  wrote  about  in  question  1  work?  Explain.

d. How  did  you  represent  the  information  in  the  problem?  Did  you  draw  a  picture,  make  a  table,  or  create  some  other  representation?  Did  you  keep  the  same  representation  as  you  worked  on  the  problem  or  did  you  change  representations?

4. Think  about  how  you  and  your  partner  figured  out  whether  you  were  making  progress.

a. Did  you  stop  and  think  about  whether  or  not  what  you  were  doing  was  working?  How?

b. If  what  you  were  doing  wasn’t  working,  did  you  talk  about  other  things  you  could  try  instead?

5. Did  you  use  the  Mathematical  Problem-­‐Solving  Routine  from  the  unit  Getting  started  with  algebra?    If  so,  how  did  you  use  it? Problem  solving  and  metacognition   3

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

LESSON 1: REVIEW ONLINE ASSESSMENT

You  will  work  with  your  class  to  review  the  online  assessment  questions.

Problems  we  did  well  on:   Skills  and/or  concepts  that  are  addressed  in  the  problems  we  did  well  on:

Problems  we  did  not  do  well  on:   Skills  and/or  concepts  that  are  addressed  in  the  problems  we  did  not  do  well  on:

Use  this  page  and  notebook  paper  to  take  notes  and  re-­‐work  particular  online  assessment  problems  that  your  class  identifies.

Problem  #_____

Work  for  problem:

Problem  #_____

Work  for  problem:

Problem  #_____

Work  for  problem: 4   Problem  solving  and  metacognition

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

LESSON 1: HOMEWORK Notes  or  additional  instructions  based  on  whole-­‐class  discussion  of  homework  assignment:

Next  class  period,  you  will  take  a  mid-­‐unit  assessment.  One  good  study  skill  to  prepare  for  tests  is  to  review  the  important  skills  and  ideas  you  have  learned.  Use  this  list  to  help  you  review  these  skills  and  concepts,  especially  by  reviewing  related  course  materials.

Important  skills  and  ideas  you  have  learned  so  far  in  the  unit  Introduction  to  functions  and  equations:

• Use  words,  tables,  graphs,  and  algebraic  rules  to  identify,  describe,  and  analyze  patterns  and  mathematical  relationships    • Solve  problems  and  model  situations  using  patterns  and  mathematical  relationships    • Connect  different  representations  of  mathematical  relationships    • Understand  the  advantages  and  limitations  of  particular  representations  of  mathematical  relationships  • Analyze  and  create  equivalent  algebraic  expressions  and  rules  • Understand  ideas  related  to  “allowable  inputs,”  discrete  versus  continuous  data,  and  proportional  versus  non-­‐proportional

linear  relationships

Homework  Assignment

Part  I:   Study  for  the  mid-­‐unit  assessment  by  reviewing  the  key  ideas  listed  above.

Part  II:   Complete  the  online  More  practice  in  the  topic  Representing  mathematical  relationships  in  multiple  ways.  Note  the  skills  and  ideas  for  which  you  need  more  review,  and  refer  back  to  related  activities  and  animations  from  this  topic  to  help  you  study.

Part  III:   Complete  Lesson  1:  Staying  Sharp. Problem  solving  and  metacognition   5

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

LESSON 1: STAYING SHARP Practic

ing  skills  &  con

cepts

1. If  Michelle  drove  273  miles  in  4  hours,  what  was  her  speed  in  miles  per  hour?    Answer  with  supporting  work:

2. Consider  the  circles  in  this  diagram:

Prep

aring  for  u

pcom

ing  lesson

s

3. Complete  the  table  for  this  magic  number  puzzle:

Directions   Example  For  any  number,

n

Choose  a  number

8   n

5  less  than  the  number

Multiply  by  3

4. Matchsticks  are  used  to  make  the  pattern  shown.  To  make  these  4  squares,  you  need  13  matchsticks.  How  many  matchsticks  do  you  need  to  make  20  squares?

Review

ing  ideas  from

earlier  g

s

5. These  statements  involve  exponents.  Complete  the  blanks  to  make  each  statement  true.

32  =  2  ·∙  2  ·∙  2  ·∙  2  ·∙  2  =  2  ____

____  =  2  ·∙  2  =  2  ____

2  =  2  ____

____  =  20

6. Anna  has  2  pairs  of  pants  (brown  and  gray)  and  3  shirts  (pink,  lavender,  and  yellow).  List  the  different  outfits  she  can  make  with  one  pair  of  pants  and  one  shirt.  (Hint:  It  may  be  faster  to  list  if  you  abbreviate.) 6   Problem  solving  and  metacognition

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc. Problem  solving  and  metacognition   7

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

Lesson 2: Checking for understanding

LESSON 2: OPENER Complete  the  table  to  show  different  ways  of  representing  the  relationship  between  the  number  of  pentagonal  tables  that  are  pushed  together  and  the  number  of  people  that  can  be  seated.    Picture  or  diagram

1  pentagonal  table

2  pentagonal  tables  pushed  together

3  pentagonal  tables  pushed  together

Table

Number  of    tables,  n

Number  of  people  seated,  p

Words    Graph

Algebraic  rule

LESSON 2: MID-UNIT ASSESSMENT

Today  you  will  take  a  mid-­‐unit  assessment. 8   Problem  solving  and  metacognition

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

LESSON 2: CONSOLIDATION ACTIVITY      Recall  Matthew,  the  baseball  player  from  earlier  in  the  course.  When  he  graphed  data  from  a  recent  baseball  drill,  he  did  not  get  the  graph  he  was  expecting.

.

When  making  a  graph,  the  choice  of  scales  to  use  for  the  axes  is  critical.  It  determines  whether  or  not  the  graph  will  accurately  depict  a  mathematical  situation  or  relationship.  In  this  topic,  you  will  practice  scaling  graph  axes.  A  learning  goal  is  that  you  will  move  closer  to  automaticity  with  this  important  skill.      Graph  the  data  for  Matthew’s  recent  baseball  drill  using  the  grid  provided.  Choose  an  appropriate  scale  for  both  the  x-­‐axis  and  the  y-­‐axis.  Your  choice  of  scales  should  allow  you  to  see  a  good  representation  of  the  data.  The  sketch  shown  (which  does  not  contain  numbers  on  the  axes)  is  provided  as  a  guide  for  what  your  finished  graph  should  look  like.

x

y

Time  from  beginning  of  throw  (seconds)

Heigh

t  of  b

all  from  groun

d  (fee

t) Problem  solving  and  metacognition   9

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc. 10   Problem  solving  and  metacognition

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

LESSON 2: HOMEWORK Notes  or  additional  instructions  based  on  whole-­‐class  discussion  of  homework  assignment:

Part  I.  Complete  the  Consolidation  activity  if  you  did  not  have  time  to  do  so  in  class.

Part  II.  A  critical  step  in  creating  a  graph  is  deciding  how  to  scale  the  axes.  In  this  assignment,  you  will  practice  this  skill.

1. State  an  appropriate  scale  to  use  to  graph  the  data  in  the  x-­‐y  table  shown:

x   −5   −4   −3   −2   −1   0   1   2   3   4   5

y   5   4   3   2   1   0   −1   −2   −3   −4   −5

Minimum  x-­‐value:     Minimum  y-­‐value:   Maximum  x-­‐value:     Maximum  y-­‐value:   Increment  for  x-­‐axis:     Increment  for  y-­‐axis:

2. State  an  appropriate  scale  to  use  to  graph  the  algebraic  rule   2 3y x= − on  the  graphing  calculator.  Consider  inputs  from  10x = − to   10x = .  You  might  find  it  helpful  to  create  an  x-­‐y  table  for  the  algebraic  rule  to  assist  you  in  making  decisions

3. Data  for  a  car  traveling  on  a  highway  are  contained  in  the  table.  State  an  appropriate  scale  to  use  to  graph  the  data.

Time  (hrs)   0   0.5   1   1.5   2   2.5   3   3.5   4

Distance  (miles)   0   30   60   90   120   150   180   210   240

Minimum  x-­‐value:     Minimum  y-­‐value:   Maximum  x-­‐value:     Maximum  y-­‐value:   Increment  for  x-­‐axis:     Increment  for  y-­‐axis: Problem  solving  and  metacognition   11

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

4. An  input-­‐output  table  for  the  algebraic  rule   2y x= is  provided.  State  an  appropriate  scale  to  use  to  graph  the  algebraic  rule.

x   −5   −4   −3   −2   −1   0   1   2   3   4   5

y   25   16   9   4   1   0   1   4   9   16   25

5. In  the  study  of  electricity,  the  relationship  between  power,  P,  (measured  in  watts),  current,  I,  (measured  in  amperes),  and

voltage,  V,  (measured  in  volts)  is  given  by  the  formula:   =P I V⋅ .  Graph  the  relationship  between  power  and  current  for  a  12-­‐volt  battery.  In  other  words,  the  goal  is  to  consider  what  graph  scale  would  make  sense  for  the  algebraic  rule   =P I 12⋅ .  Consider  inputs  for  the  variable  I from  0  to  10  amperes.  State  an  appropriate  scale  to  use  for  the  graph.  You  might  find  it  helpful  to  create  an  input-­‐output  table  for  the  algebraic  rule  to  assist  you  in  making  decisions  about  the  graph  scales.

Minimum  x-­‐value:     Minimum  y-­‐value:   Maximum  x-­‐value:     Maximum  y-­‐value:   Increment  for  x-­‐axis:     Increment  for  y-­‐axis: 12   Problem  solving  and  metacognition

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

LESSON 2: STAYING SHARP Practic

ing  skills  &  con

cepts

1. It  took  Marty  10  seconds  to  walk  15  feet.  How  fast  was  he  walking,  in  feet  per  second?    Answer  with  supporting  work:

2. A  particular  shade  of  purple  paint  is  made  of  a  ratio  of  2  pints  of  red  paint  to  1  pint  of  blue  paint.  List  4  other  combinations  of  red  and  blue  paint  that  would  make  the  same  shade  of  purple  paint.

Prep

aring  for  u

pcom

ing  lesson

s

3. Complete  the  table  for  this  magic  number  puzzle:

Directions   Example  For  any  number,

n

Choose  a  number   7   n

Multiply  by  3

4. Matchsticks  are  used  to  make  the  pattern  shown.  To  make  these  4  squares,  you  need  13  matchsticks.  How  many  squares  in  this  pattern  could  you  make  with  55  matchsticks?

Review

ing  ideas  from

earlier  g

s

5. These  statements  involve  exponents.  Complete  the  blanks  to  make  each  statement  true.

27  =  3  ·∙  3  ·∙  3  =  3  ____

____  =  3  ·∙  3  =  3  ____

3  =  3  ____

____  =  30

6. It's  peanut  butter  and  jelly  time!  Fred  has  two  types  of  peanut  butter  (smooth  and  crunchy),  three  types  of  jelly  (raspberry,  grape,  and  blueberry),  and  one  type  of  bread  (wheat).  How  many  different  peanut  butter  and  jelly  sandwiches  can  he  make?    Answer  with  justification: Problem  solving  and  metacognition   13

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

Lesson 3: Metacognition and the Friendship Club Problem

LESSON 3: OPENER Mike  and  Ike  are  working  on  the  following  problem:  Imagine  that  a  cell  splits  itself  into  two  cells  every  24  hours.  How  many  cells  will  there  be  after  5  days?

1. Consider  Mike’s  plan  for  solving  the  problem.  Is  he  on  the  right  track  to  finding  a  solution?  Explain.    Mike’s  work

If  the  cell  splits  each  day,  then  there  will  be  2  cells  on  the  second  day,  3  cells  on  the  next  day,  and  so  on.  I  can  use  a  table  to  help  me  keep  track  of  the  situation.

Day   1   2   3  Number  of  cells   1   2   3

2. Consider  Ike’s  plan  for  solving  the  problem.  Is  he  on  the  right  track  to  finding  a  solution?  Explain.

3. When  solving  a  problem,  why  is  it  a  good  idea  to  check  to  make  sure  that  you  are  on  the  right  track?

4. If  you  realize  that  you  are  not  on  the  right  track,  what  can  you  do?

Ike’s  work

This  diagram  shows  how  the  cell  splits  over  the  first  3  days.

Day  1   Day  2   Day  3 14   Problem  solving  and  metacognition

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

LESSON 3: CORE ACTIVITY The  components  of  the  Mathematical  Problem-­‐Solving  Routine  are  shown.  How  are  the  metacognitive  questions  related  to  the  problem-­‐solving  routine?

Mathematical  Problem  Solving  Routine   Metacognitive  questions

• How  is  this  problem  similar  to  other  problems  I  have  solved?

• What  questions  do  I  have  about  the  problem?  • What  are  some  entry  points  into  this  problem?  • How  can  I  represent  this  problem?

• What  tools  do  I  have  available  to  help  me  with  this  task?

• Which  strategies  might  help  me?  • What  should  I  do  first?

• Am  I  on  the  right  track?  • What  should  I  do  next?  • Should  I  try  a  different  approach?  • If  I’m  stuck,  what  other  strategies  can  I  try?

• Does  this  answer  make  sense  in  the  context  of  the  problem?

• How  close  is  this  answer  to  my  estimate?  • Are  there  things  I  still  don’t  understand?  • What  did  I  learn  from  this  problem  that  I  could

use  in  solving  other  problems? Problem  solving  and  metacognition   15

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

1. For  each  step  in  the  Mathematical  Problem-­‐Solving  routine,  list  additional  metacognitive  questions  that  you  can  ask  yourself.

a.  Understand

b.  Make  a  plan

c.  Do  the  math

d.  Look  back

2. How  can  metacognitive  strategies  help  you  be  a  more  effective  learner? 16   Problem  solving  and  metacognition

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

3. The  Friendship  Club  Problem:  You  are  a  member  of  a  friendship  club.  There  are  8  members  in  your  club.  On  the  8th  day  of  each  month,  you  have  a  friendship  call.  Each  member  of  the  club  talks  to  every  other  member  that  day  by  phone.  This  way  everybody  expresses  his  or  her  friendship  with  each  other.

a. How  many  conversations  occur  on  the  8th  day  of  each  month?

b. How  many  conversations  occur  each  year?  Show  how  you  figured  this  out.

c. If  the  club  adds  four  new  friends,  how  many  conversations  will  occur  each  month?  Explain  your  solution.

d. Another  friendship  club  likes  your  friendship  conversation  idea.  The  members  of  the  other  club  want  to  know  how  they  can  figure  out  how  many  conversations  will  occur  given  any  number  of  members.  Explain  how  they  can  figure  out  the  total  number  of  conversations  for  each  month. Problem  solving  and  metacognition   17

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

LESSON 3: CONSOLIDATION ACTIVITY

1. Uri  and  Susan  are  lab  partners  in  science  class.  They  collect  data  (shown  below)  on  the  heating  curve  of  water.

Time  (sec)   0   10   20   30   40   50   60   70   80   90   100   110   120   130   140   150   160   170   180

Temp  (°C)   −25   −10   0   0   0   0   0   15   28   43   58   73   88   100   100   100   100   100   100    They  want  to  create  a  graph  of  the  data  on  the  graphing  calculator.  State  an  appropriate  scale  for  the  axes  for  the  graph.

2. You  want  to  graph  the  algebraic  rule   3 5y x= + .  Suppose  you  want  to  consider  both  positive  and  negative  values  for  the

input.  State  an  appropriate  scale  for  the  axes  for  the  graph.  You  might  find  it  helpful  to  create  an  x-­‐y  table  for  the  algebraic  rule  to  assist  you  in  making  decisions  about  the  graph  scales.

Minimum  x-­‐value:     Minimum  y-­‐value:   Maximum  x-­‐value:     Maximum  y-­‐value:   Increment  for  x-­‐axis:     Increment  for  y-­‐axis:

3. Noemi  is  performing  a  walking  experiment  in  her  science  class  using  a  motion  detector.  She  walks  20  feet  from  the  motion

detector;  this  part  of  the  walk  takes  10  seconds.  She  stops  for  5  seconds.  Finally,  she  walks  back  to  the  motion  detector;  this  part  of  the  walk  takes  15  seconds.  Noemi  needs  to  create  a  graph  of  the  walk.  Time,  in  seconds,  will  be  graphed  on  the  x-­‐axis.  Distance  from  the  motion  detector,  in  feet,  will  be  graphed  on  the  y-­‐axis.  State  an  appropriate  scale  for  the  axes  for  the  graph.

Minimum  x-­‐value:     Minimum  y-­‐value:   Maximum  x-­‐value:     Maximum  y-­‐value:   Increment  for  x-­‐axis:     Increment  for  y-­‐axis:

4. You  want  to  graph  the  function   2y x= using  your  graphing  calculator.  State  an  appropriate  scale  for  the  axes  for  the  graph.

Consider  inputs  from  10  to  −10.  You  might  find  it  helpful  to  create  an  x-­‐y  table  for  the  algebraic  rule  to  assist  you  in  making  decisions  about  the  graph  scales.

5. The  formula   1.8 32F C= + describes  the  relationship  between  the  Fahrenheit  and  Celsius  temperature  scales.  Suppose  you

want  to  create  a  graph  that  shows  the  relationship  for  Celsius  temperatures  from  0  to  100  degrees.  State  an  appropriate  scale  for  the  axes  for  the  graph.  You  might  find  it  helpful  to  create  an  input-­‐output  table  for  the  algebraic  rule  to  assist  you  in  making  decisions  about  the  graph  scales.

Minimum  x-­‐value:     Minimum  y-­‐value:   Maximum  x-­‐value:     Maximum  y-­‐value:   Increment  for  x-­‐axis:     Increment  for  y-­‐axis: 18   Problem  solving  and  metacognition

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

LESSON 3: HOMEWORK Notes  or  additional  instructions  based  on  whole-­‐class  discussion  of  homework  assignment:

Homework  Assignment

Part  I:   Complete  the  online  More  practice  in  the  topic  Problem  solving  and  metacognition.

Part  II:   Complete  Lesson  3:  Staying  Sharp.

Record  any  questions  that  you  have  as  you  complete  the  More  practice  (perhaps  as  it  connects  to  challenges  you  encountered  with  particular  online  assessment  questions)  below. Problem  solving  and  metacognition   19

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

LESSON 3: STAYING SHARP Practic

ing  skills  &  con

cepts

1. It  took  Mary  15  seconds  to  walk  10  feet.  How  fast  was  she  walking,  in  feet  per  second?    Answer  with  supporting  work:

2. Find  the  ratio  of  width  to  length  for  each  rectangle.  Simplify  each  ratio.

Width   Length   Ratio

2   8

6   24

5   20

3   12

Prep

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3. In  an  algebra  class  last  year,  students  who  earned  grades  of  A  spent  4.7  hours  more  on  homework  per  week  than  students  who  earned  grades  of  C.  Write  an  algebraic  rule  to  represent  the  situation,  using  A  for  the  hours  spent  on  homework  per  week  for  students  who  earned  grades  of  A,  and  C  for  the  hours  spent  on  homework  per  week  for  students  who  earned  a  grade  of  C.    Answer  with  supporting  work:

4. Kory  bought  2  boxes  of  candy  and  1  soda  at  the  movies  and  paid  \$8.00.  Christopher  bought  1  box  of  candy  and  1  soda  and  paid  \$5.50.  How  much  does  1  box  of  candy  cost?  How  much  does  1  soda  cost?    Answer  with  justification:

Review

ing  ideas  from

earlier  g

s

5. These  statements  involve  exponents.  Complete  the  blanks  to  make  each  statement  true.

4  ·∙  4  ·∙  4  ·∙  4  =  256  =  4  ____

4  ·∙  4  ·∙  4  =  ___  =  4  ____

4  ·∙  4  =  ___  =  4  ____

4  =  4  ____

6. Three  soccer  players,  Alberto,  Ben,  and  Chike,  are  forming  a  "wall"  to  defend  against  a  penalty  kick.  How  many  different  ways  can  they  line  up  side  by  side?

Make  a  list  and  explain  how  you  know  you  found  all  of  the  possible  arrangements. 20   Problem  solving  and  metacognition

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc. Problem  solving  and  metacognition   21

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

Lesson 4: Friendship Club Problem presentations

LESSON 4: OPENER In  this  lesson,  you  and  your  partner  will  prepare  and  present  a  solution  to  the  Friendship  Club  Problem.  Review  the  Class  Presentation  Criteria  then  answer  the  following  question.

Class  Presentation  Criteria  Speakers   Audience  Members

1. Include  a  clear  write-­‐up  of  your  solution  to  the  problem.

2. Include  a  clear,  concise  explanation  as  to  why  you  believe  your  answer  is  correct.

3. Include  a  clear,  concise  explanation  of  solution  strategy.

4. Both  partners  participate  in  presentation.

5. Both  partners  use  strong,  clear  voices  when  making  the  presentation;  both  partners  employ  good  posture  and  make  eye  contact.

1. Give  their  full  attention  and  respect  to  the  presenter.

2. Take  notes,  as  needed.

3. Are  prepared  to  summarize  the  presenters’  mathematical  argument  or  strategy.

5. Are  prepared  to  make  connections  between  the  presenters’  ideas  and  their  own  ideas.

1. What  is  a  goal  you  have  for  the  presentation  that  you  will  make  to  the  class?

LESSON 4: CORE ACTIVITY 1. Work  with  your  partner  to  prepare  a  write-­‐up  of  the  solution  to  the  Friendship  Club  Problem  along  with  a  clear,  concise

As  different  solutions  are  presented,  note  how  different  representations  were  used  to  solve  the  problem.  Also  note  different  ways  in  which  the  Mathematical  Problem-­‐Solving  Routine  and  metacognitive  strategies  were  helpful  in  solving  the  problem.

My  notes:

2. Use  the  table  to  find  the  number  of  calls  each  member  of  the  Friendship  Club  makes.  Then  use  the  completed  table  to  find  the  general  rule  showing  the  total  number  of  calls  a  Friendship  Club  will  make,  given  any  number  of  members.

3. In  the  last  lesson,  you  wrote  a  general  rule  to  find  the  number  of  calls  a  Friendship  Club  will  make  given  any  number  of

members.  How  was  the  method  you  used  to  find  this  rule  similar  to  or  different  from  the  method  presented  in  the  animation?

LESSON 4: REVIEW MID-UNIT ASSESSMENT

Person   Number  of  calls

1   n  –  1

2   3   4

n  –  1   n 22   Problem  solving  and  metacognition

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

Use  the  Assessment-­‐Processing  Routine  to  help  review  and  correct  your  mid-­‐unit  assessment.

LESSON 4: HOMEWORK Notes  or  additional  instructions  based  on  whole-­‐class  discussion  of  homework  assignment:

1. Complete  the  following  math  journal.

Term   My  understanding  of  what  the  idea  means   An  example  that  shows  the  meaning  of  the  idea

Metacognition

2. You  presented  the  Friendship  Club  Problem  in  class  today.  What  are  some  ways  that  you  used  the  metacognitive  questions  to  solve  this  problem? Problem  solving  and  metacognition   23

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

Ashley  and  Brittany's  Trip  Problem

Ashley  and  Brittany  are  cousins  who  are  going  on  a  trip  with  their  families  to  visit  their  grandparents  in  Mississippi.  The  families  decide  to  take  two  cars  on  the  trip,  but  both  families  leave  from  Ashley’s  house  and  take  the  same  route.  Ashley’s  family  drives  in  one  car  at  a  steady  rate  of  40  miles  per  hour.  Brittany’s  family  leaves  two  hours  later  and  drives  at  a  constant  rate  of  50  miles  per  hour.  Both  families  arrive  at  the  grandparents’  house  at  the  same  time!  What  is  the  distance  from  Ashley’s  house  to  the  grandparents’  house?

3. Use  the  Mathematical  Problem-­‐Solving  Routine  to  solve  the  problem  of  Ashley  and  Brittany's  Trip.  The  steps  are  outlined  below.  As  you  apply  the  Problem-­‐Solving  Routine  to  this  problem,  make  note  of  where  you  are  also  using  metacognitive  strategies.  The  last  question  asks  you  to  write  the  ways  you  used  metacognitive  strategies.

Understand

a. Try  to  visualize  the  situation.  Consider  drawing  a  diagram  to  help  you  picture  the  situation.

b. State  the  goal  of  the  problem  in  your  own  words.

c. What  is  the  important  information  in  the  problem?

Plan

d. What  strategy/representation  will  you  try  to  solve  the  problem?  Why?

Solve

Look  back

f. Is  the  answer  that  you  got  reasonable?

h. What  can  you  learn  from  this  problem?  What  strategy  did  you  use  and  why  did  it  work  for  this  problem?  For  what  types  of  problems  would  this  strategy  work  in  general?

Metacognitive  strategies

i. When  you  worked  Ashley  and  Brittany's  Trip  Problem,  what  are  some  ways  that  you  used  the  metacognitive  questions? 24   Problem  solving  and  metacognition

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

LESSON 4: STAYING SHARP Practic

ing  skills  &  con

cepts

1. Johann  walked  1.2  feet  per  second  How  far,  in  feet,  did  he  walk  in  5  seconds?    Answer  with  supporting  work:

2. Two  boxes  of  identical  birthday  candles  are  pictured:  a  2-­‐ounce  box  and  a  6-­‐ounce  box.  There  are  96  candles  in  the  large  box.  How  many  candles  are  in  the  small  box?

Prep

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3. Two  bottles  of  cough  medicine  are  the  same  size.  The  cost  of  the  Store  Brand  is  two-­‐thirds  the  cost  of  the  National  Brand.  If  you  use  S  to  represent  the  cost  of  the  Store  Brand  and  N  to  represent  the  cost  of  the  National  Brand,  which  of  these  equations  correctly  represents  the  situation?

3S N=   23S N=

2

3N S=   2N S=

4. Using  the  situation  in  question  3,  if  one  bottle  of  the  National  Brand  of  cough  medicine  costs  \$12,  what  is  the  cost  of  a  bottle  of  the  Store  Brand?    Answer  with  justification:

Review

ing  ideas  from

earlier  g

s

5. These  statements  involve  exponents.  Complete  the  blanks  to  make  each  statement  true.    54  =  ______

53  =  ______

52  =  ______

51  =  ______    50  =  ______

6. There  are  6  different  ways  to  arrange  the  three  letters  A,  B,  and  C:  ABC,  ACB,  BAC,  BCA,  CAB,  CBA.    There  are  24  different  ways  to  arrange  the  four  letters  A,  B,  C,  and  D.  The  table  below  contains  a  list  of  these  arrangements  in  a  systematic  order.  Complete  the  table.    ABCD   BACD   CABD   DABC  ABDC   BA___   C____   D____  ACBD   BCAD   CB___   DB___  AC___   BC___   CB___   D____  ADBC   BDAC   CDAB   D____  AD___   BD___   CD___   _____ Problem  solving  and  metacognition   25

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

Lesson 5: Solving a similar problem

LESSON 5: OPENER Of  the  three  problems  below,  which  two  do  you  think  are  the  most  mathematically  similar?  You  do  not  have  to  solve  the  problems,  but  you  should  provide  an  explanation  for  your  selection.    Problem  1

Find  the  length  of  the  hypotenuse  of  the  right  triangle  shown.

Problem  2

On  a  baseball  diamond,  the  distance  between  each  of  the  bases  is  90  feet,  as  shown  in  the  diagram.  A  player  hits  a  home  run  and  travels  around  the  bases  (from  home  plate  to  first  base  to  second  base  to  third  base  to  home  plate).  What  is  the  total  distance  around  the  bases?

Problem  3

On  a  baseball  diamond,  the  distance  between  each  of  the  bases  is  90  feet,  as  shown  in  the  diagram.  A  player  throws  the  ball  from  home  plate  to  second  base.  What  is  the  distance  from  home  plate  to  second  base?      Answer:    Problems            and      are  mathematically  similar  because  …

LESSON 5: CORE ACTIVITY 1. Examine  the  following  six  problems.  Which  problems  do  you  think  are  mathematically  alike?  Make  at  least  two  groups  of

problems  and  record  them  as  Group  1  and  Group  2.  If  you  find  more  groups,  you  can  record  them  as  Group  3  and  Group  4.  For  each  group,  write  the  reason  you  have  grouped  those  problems  together.

A. Ella  had  \$40.  She  spent  \$15  on  music.  How  much  does  she  have  left?

B. North  High  School  is  having  a  charity  sale  in  the  gymnasium.  If  750  of  the  gym’s  7,000  sq  ft  of  floor  space  is  used  to  sell  refreshments,  how  much  floor  space  is  available  for  selling  merchandise?

C. Mike’s  bedroom  is  9  ft  by  12  ft.  How  many  square  feet  of  floor  space  does  he  have?

D. A  teacher  bought  5  new  storage  shelves.  Each  shelf  holds  45  student  journals.  How  many  journals  can  she  store  on  her  new  shelves?

E. Jan  earns  \$15  per  hour  tutoring  math.  How  much  will  she  earn  if  she  tutors  for  40  hours?

F. 16  of  the  25  best-­‐selling  albums  of  all  time  are  by  groups.  How  many  are  not  by  groups? 26   Problem  solving  and  metacognition

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

Group  1:

Reason:

Group  2:

Reason:

Group  3  (if  needed):

Reason:

Group  4  (if  needed):

Reason:

2. List  ways  in  which  the  Frozen  Yogurt  Store  Problem  and  the  Friendship  Club  Problem  are  mathematically  alike  and  ways  in

which  they  are  different.

Alike:

Different:

3. Apply  the  Mathematical  Problem-­‐Solving  Routine  and  metacognitive  strategies  to  solve  the  Diagonals  Problem.

The  table  shows  the  relationship  between  the  number  of  sides  of  a  polygon  and  the  number  of  diagonals  that  can  be  drawn  in  the  polygon.  (Remember:  A  diagonal  is  a  line  segment  inside  a  polygon  that  joins  two  vertices  of  the  polygon.  A  diagonal  cannot  be  an  edge  of  a  polygon.)

a. Based  on  this  pattern,  how  many  diagonals  would  a  7-­‐sided  polygon  have?  Justify  your  answer. Problem  solving  and  metacognition   27

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

b. Draw  the  diagonals  on  the  7-­‐sided  polygon  shown.

c. How  many  diagonals  does  a  polygon  with  20  sides  have?  Justify  your  answer.

d. How  many  diagonals  does  a  polygon  with  any  number  of  sides,  n,  have?  Justify  your  answer.

e. If  a  polygon  has  65  diagonals,  how  many  sides  does  that  polygon  have?  Justify  your  answer.

f. What  are  some  of  the  ways  you  used  the  Mathematical  Problem-­‐Solving  Routine  and  metacognitive  questions  when  you  worked  on  this  problem?

g. Is  this  problem  similar  to  a  problem  that  you  have  solved  before?  If  so,  name  the  problem(s)  and  tell  how  these  problems  are  similar.  Did  the  similarity  help  you  solve  the  problem?  If  so,  how?

LESSON 5: CONSOLIDATION ACTIVITY 1. What  are  some  of  the  ways  that  you  used  the  Mathematical  Problem-­‐Solving  Routine  and  metacognitive  questions  when

you  worked  on  the  problems  in  this  topic:  the  Frozen  Yogurt  Store  Problem,  the  Friendship  Club  Problem,  Ashley  and  Brittany’s  Trip  Problem,  and  the  Diagonals  Problem? 28   Problem  solving  and  metacognition

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

2. What  benefits  did  you  experience  from  using  the  Mathematical  Problem-­‐Solving  Routine  and  these  metacognitive  strategies?

3. Think  about  the  problem-­‐solving  strategies  you  used  to  solve  the  Frozen  Yogurt  Store  Problem,  the  Friendship  Club  Problem,  Ashley  and  Brittany’s  Trip  Problem,  and  the  Diagonals  Problem.  Record  the  strategies  in  your  Personal  Record  Sheet  on  the  appropriate  line.

As  you  continue  in  this  course,  you  will  keep  adding  problem-­‐solving  strategies  to  your  math  toolbox.  The  more  tools  you  have,  the  better  equipped  you  will  be  to  tackle  and  solve  challenging  problems. Problem  solving  and  metacognition   29

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

Personal  Record  Sheet:  Problem-­‐Solving  Strategies  Mark  a  ✓  for  strategies  that  you  used  to  work  on  the  problem  or  problem  type.

Problem  or  Problem  Type

Strategies  Used

Solve  a  similar

prob

lem

Draw

a  diagram

Make  a  table

Make  a  grap

h

Make  an

organized  list

Solve  a  simpler

prob

lem

Look  fo

r  a  pattern

Set  u

p  an

d  solve

an  equ

ation

Gue

ss-­‐and

-­‐che

ck

Use  logical

reason

ing

Work  ba

ckwards

Act  it  o

ut

The  Frozen  Yogurt  Store  Problem

The  Friendship  Club  Problem

Ashley  and  Brittany’s  Trip  Problem

Diagonals  Problem 30   Problem  solving  and  metacognition

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

LESSON 5: HOMEWORK Notes  or  additional  instructions  based  on  whole-­‐class  discussion  of  homework  assignment:

The  Will

The  family  of  Mr.  I.  M.  Rich,  an  eccentric  millionaire,  is  gathered  in  his  lawyer’s  office  for  the  reading  of  his  will.  The  lawyer  reads  the  following:

I,  I.M.  Rich,  distribute  my  fortune  as  follows  and  in  this  order:

• The  first  \$5  million  goes  to  my  best  friend,  Sam,  my  dog;

• Half  the  remainder  goes  to  my  high  school  algebra  teacher;  because  of  her,  I  learned  the  math  I  needed  to  make  my  fortune;

• Then,  I  give  \$50  million  to  build  a  new  hospital  in  town;

• Of  the  remaining  amount,  50%  goes  to  my  wife,  40%  to  my  daughter,  9%  to  my  lawyer,  and  1%  to  my  son  who  has  never  worked  a  day  in  his  life.  It’s  time  he  gets  started.

“What!”  says  the  son.  “All  I  get  is  a  measly  \$10,000!”

3. What  are  some  of  the  ways  that  you  used  the  Mathematical  Problem-­‐Solving  Routine  and  metacognitive  strategies  when

you  worked  on  this  problem?

4. Is  this  problem  similar  to  a  problem  that  you  have  solved  before?  If  so,  name  the  problem(s)  and  tell  how  these  problems  are  similar.  Did  the  similarity  help  you  solve  the  problem?  If  so,  how? Problem  solving  and  metacognition   31

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.

LESSON 5: STAYING SHARP Practic

ing  skills  &  con

cepts

1. Luis  has  two  cars.  His  minivan  gets  18  miles  per  gallon  of  gasoline  and  his  compact  gets  24  miles  per  gallon.  How  many  gallons  of  gasoline  will  he  save  if  he  travels  216  miles  using  the  compact  instead  of  the  minivan?    Answer  with  supporting  work:

2. To  get  ready  for  a  field  trip,  students  and  adults  are  put  into  groups.  For  every  12  students  in  a  group,  there  are    2  adults.  If  there  are  96  students  on  the  field  trip,  how  many  adults  are  on  the  field  trip?    Answer  with  justification:

Prep

aring  for  u

pcom

ing  lesson

s

3. Casey,  a  second-­‐grade  boy,  is  raising  caterpillars  which  each  have  the  same  number  of  legs.  Let  C  represent  the  number  of  caterpillars  Casey  has,  and  L  represent  the  number  of  legs  each  caterpillar  has.  Write  an  expression  for  the  total  number  of  legs  that  the  caterpillars  and  Casey  have  together.    Answer  with  supporting  work:

4. Complete  the  square  box  problem.

Review

ing  ideas  from

earlier  g

s

5. These  statements  involve  exponents.  Complete  the  blanks  to  make  each  statement  true.    20  =  _____

2  ____  =  2

2  ____    =  32

26  =  _____

6. This  table  shows  the  number  of  possible  arrangements  of  n  letters.  For  example,  there  are  3  ·∙  2  ·∙  1  =  6  different  ways  to  arrange  3  letters.  Complete  the  table.    Number  of  letters   Process

Number  of  arrangements

1   1   1

2   2  ·∙  1   2

3   3  ·∙  2  ·∙  1   6

4

5

6

-2

-26 32   Problem  solving  and  metacognition

Copyright  ©  2016  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin,  Learning  Sciences  Research  Institute  at  the  University  of  Illinois  at  Chicago,  Agile  Mind,  Inc.