Best Multiplication Unit Ever

 

 

 

 

 

 

 

 

 


 
Multiplication:
 Just
 the
 Facts,
 Ma’am
 

Mary
 Ebejer
 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

EDG
 630
 -­‐
 01
 
Teaching
 Mathematics
 K-­‐8
 

November
 30,
 2010
 


 

 

Table
 of
 Contents
 

Introduction
  ................................................................................................................................
 3
 
Unit
 Standards
 (GLCEs)
 ................................................................................................................
 3
 
“Big
 Ideas”
  .................................................................................................................................... 4
 
Assessments
  ................................................................................................................................. 5
 
Lessons:
 
 
 
  1:
 
 
  Multiplication
 Illustration
 ............................................................................ 6
 

  2:
 
 
  Fish
 Bowl ....................................................................................................... 8
 

  3:
 
 
  Groupings
 All
 Around
 Us
  ........................................................................... 11
 

  4:
 
  Game:
 Circle
 and
 Stars
 ............................................................................... 17
 

  5:
 
 
  Creating
 Multiplication
 Tables
  .................................................................
 20
 

  6:
 
  Billy
 Wins
 a
 Shopping
 Spree!
 ..................................................................... 29
 

 
References
 .................................................................................................................................. 31
 


 

2

 

 

Introduction
 


 
This
 third
 and
 fourth
 grade
 math
 unit
 explores
 the
 various
 meanings
 and
 representations
 of
 
multiplication
 as
 “repeated
 addition.
 The
 lessons
 in
 this
 unit
 rely
 on
 extensive
 use
 of
 
manipulatives,
 as
 well
 as
 math
 songs
 and
 games
 and
 art
 activities
 to
 help
 students
 identify
 
situations
 when
 multiplication
 would
 be
 useful,
 to
 reinforce
 their
 learning
 and
 to
 improve
 
recall
 speed
 for
 multiplication
 facts.
 

 
Many
 of
 the
 activities
 in
 the
 unit
 involve
 cooperative
 learning
 in
 pairs,
 small
 groups
 and
 as
 a
 class
 
as
 a
 whole.
 Students
 should
 see
 their
 classroom
 as
 a
 place
 where
 cooperation
 and
 collaboration
 
are
 valued
 and
 expected.
 It
 respects
 the
 principle
 that
 interaction
 fosters
 learning
 and
 that
 
cooperative
 group
 work
 is
 basic
 the
 classroom
 culture.
 


 


 

 

 

Unit
 Standards
 

3
 
 Multiply
 and
 divide
 whole
 numbers
 
 
3.N.MR.03.09
 
 Use
 multiplication
 and
 division
 fact
 families
 to
 understand
 the
 inverse
 relationship
 
of
 these
 two
 operations,
 e.g.,
 because
 3
 x
 8
 =
 24,
 we
 know
 that
 24
 ÷
 8
 =
 3
 or
 24
 ÷
 3
 =
 8;
 express
 a
 
multiplication
 statement
 as
 an
 equivalent
 division
 statement.
 
 

 
3.N.MR.03.10
 
 Recognize
 situations
 that
 can
 be
 solved
 using
 multiplication
 and
 division
 including
 
finding
 "How
 many
 groups?"
 and
 "How
 many
 in
 a
 group?"
 and
 write
 mathematical
 statements
 to
 
represent
 those
 situations.
 
 

 
3.N.FL.03.11
 
 Find
 products
 fluently
 up
 to
 10
 x
 10;
 find
 related
 quotients
 using
 multiplication
 and
 
division
 relationships.
 
 

 
3.N.MR.03.12
 
 Find
 solutions
 to
 open
 sentences,
 such
 as
 7
 x
 __
 =
 42
 or
 12
 ÷
 __
 =
 4,
 using
 the
 
inverse
 relationship
 between
 multiplication
 and
 division.
 
 

3

 

 

“Big
 Ideas”
 

Lesson
 1:
 
 
 
  Multiplication
 Illustration
 

 
 
  Kick-­‐off
 lesson
 using
 M&M’s.
 

Lesson
 2:
 
 
 
  Fish
 Bowl
 
Multiplication
 is
 repeated
 addition.
 
 
 

Lesson
 3:
 
 
 
  Groupings
 All
 Around
 Us
 
Multiplication
 is
 a
 quick
 way
 to
 figure
 out
 how
 many
 you
 have
 
altogether
 of
 something
 when
 things
 come
 in
 groups.
 

Lesson
 4:
 
 
 
  Circle
 and
 Stars
 Game
 

 
Students
 see
 multiplication
 as
 the
 combining
 of
 equal-­‐size
 groups
 that
 
can
 be
 represented
 with
 a
 multiplication
 equation.
 

Lesson
 5:
 
 
 
  Creating
 Multiplication
 Tables
 

  Students
 create
 personal
 laminated
 multiplication
 tables
 as
 they
 

learn
 to
 recognize
 both
 the
 geometry
 and
 patterns
 inherent
 in
 

multiplication.
 

Lesson
 6:
 
 
 
  Billy
 Wins
 a
 Shopping
 Spree!
 

  We
 use
 multiplication
 everyday
 to
 solve
 real-­‐world
 problems.
 

 
Unit
 Songs:
 
  Multiplication
 songs
 
 
Introduce
 in
 morning
 playing
 a
 recording.
 During
 transition
 periods
 play
 songs.
 

  Teach
 song
 at
 end
 of
 math
 lesson
 of
 the
 day.
 Each
 student
 has
 book
 of
 songs.
 Sing
 
song
 at
 closing.
 
 
 

4

 
Unit
 Assessments
 

Teacher
 observation
 of
 class
 work,
 combined
 with
 evaluation
 of
 Student
 Portfolio
 and
 Math
 Journal
 
Entries
 to
 serve
 as
 assessments
 of
 student
 understanding
 of
 multiplication,
 both
 its
 meaning
 and
 
real-­‐world
 uses.
 
 


 
Formative:
  Periodic
 Math
 Journal
 Entries
 

Group
 Work
 and
 Class
 Discussion
 Observations
 
 
Summative:
  Student
 Portfolios
 

Final
 Math
 Journal
 Entry
 

  “What
 I
 now
 know
 about
 multiplication.”
 

5

Lesson
 1:
 
 Introduction
 to
 Multiplication
 
 

Grade
 Level
  Introduction:
 Students
 will
 discuss
 and
 write
 about
 their
 current
 understanding
 of
 
multiplication
 before
 we
 begin
 the
 unit
 of
 study.
 Recognize
 situations
 that
 can
 be
 
Third
 and
 Fourth
  solved
 using
 multiplication
 and
 division
 including
 finding
 "How
 many
 groups?"
 and
 

  "How
 many
 in
 a
 group?"
 and
 write
 mathematical
 statements
 to
 represent
 those
 

  situations.
 
Time
 Needed
 
 
Preparation:
 The
 instructor
 will
 pass
 out
 a
 journal
 to
 each
 student.
 
 The
 journal
 will
 
50
 minutes
  contain
 copies
 of
 everything
 that
 will
 be
 used
 in
 this
 unit
 including
 handouts,
 

  templates,
 and
 multiplication
 charts.
 
 The
 student
 journals
 will
 also
 contain
 blank
 paper
 

  for
 students
 to
 recorded
 their
 observations
 and
 thoughts
 as
 well
 as
 to
 use
 to
 generate
 
any
 computations
 that
 may
 be
 needed.
 
 In
 my
 classroom
 this
 journal
 is
 comprised
 of
 a
 
Materials
  two
 pocket
 folder
 that
 contains
 brads
 for
 binding
 papers.
 

 
Large
 piece
 of
  Prior
 to
 beginning
 the
 unit
 on
 multiplication,
 ask
 the
 students
 to
 respond
 to
 this
 
butcher
 paper
  prompt
 in
 their
 Math
 Journals:
 

Marker
 to
 record
 
  Write
 what
 you
 know
 about
 multiplication.
 
ideas
 
Their
 response
 will
 serve
 as
 a
 benchmark
 for
 their
 formative
 assessments
 for
 the
 unit.
 
 
A
 math
 journal
 
 
for
 each
 student
  For
 this
 particular
 lesson,
 you
 will
 need
 bags
 of
 M&M's,
 jelly
 beans,
 or
 some
 other
 
small
 candy.
 
M&M's,
 jelly
 
 
beans,
 and
 small
  GLCE:
 .N.MR.03.10
 Recognize
 situations
 that
 can
 be
 solved
 using
 multiplication
 and
 
candy
  division
 including
 finding
 "How
 many
 groups?"
 and
 "How
 many
 in
 a
 group?"
 and
 write
 
mathematical
 statements
 to
 represent
 those
 situations.
 
 

 
3.
 N.MR.03.12
 Find
 solutions
 to
 open
 sentences,
 such
 as
 7
 x
 __
 =
 42
 or
 12
 ÷
 __
 =
 4,
 

  using
 the
 inverse
 relationship
 between
 multiplication
 and
 division.
 


  N.MR.04.14
 Solve
 contextual
 problems
 involving
 whole
 number
 multiplication
 and
 
division.
 

Engagement
 (15
 minutes):
 Teacher
 led
 class
 discussion:
 “Students,
 open
 your
 math
 
journals
 to
 an
 empty
 page.
 
 As
 we
 discuss
 our
 ideas
 about
 multiplication
 you
 may
 write
 
down
 your
 thoughts,
 ideas,
 and
 observations
 in
 the
 section
 titled
 ‘What
 I
 Know’.
 
 Write
 
whatever
 you
 want
 to
 about
 the
 multiplication,
 spelling
 does
 not
 matter
 in
 this
 part.
 
 
Please
 don’t
 erase
 anything
 you
 write.
 Who’s
 ready
 to
 begin?”
 

The
 instructor
 is
 to
 ask
 a
 series
 of
 questions
 that
 follows.
 Record
 the
 answers
 on
 the
 
classroom
 KWL
 chart.
 
 
 

1.
 
 
 Has
 everyone
 heard
 about
 multiplication?
 

2.
 
 Who
 thinks
 they
 know
 what
 multiplication
 is?
 

3.
 
 Who
 thinks
 they
 could
 explain
 multiplication?
 

4.
 
 Who
 knows
 any
 multiplication
 facts?
 

5.
 
 Does
 anyone
 know
 how
 to
 solve
 a
 problem
 using
 more
 than
 one
 multiplication
 
fact?
 


 

6

Exploration
 (20
 minutes):
 M&M
 multiplication
 
Using
 real
 world
 story
 problems
 to
 solve
 multiplication
 facts
 
This
 is
 a
 lesson
 to
 help
 students
 understand
 the
 uses
 of
 multiplication
 and
 practice
 problem
 solving
 
while
 having
 fun.
 You
 will
 need
 bags
 of
 M&M's,
 jelly
 beans,
 or
 some
 other
 small
 candy.
 
 

Procedure
 
 

1)
 
 Students
 are
 divided
 into
 groups.
 
 

2)
 
 Give
 each
 group
 a
 bag
 of
 candy.
 
 

3)
 
 Explain
 that
 each
 group
 must
 share
 their
 candy
 with
 the
 other
 groups.
 
 

4)
 
 Now
 give
 each
 group
 a
 different
 problem
 to
 solve.
 For
 instance,
 if
 you
 have
 5
 groups
 with
 4
 
students
 in
 each
 group
 tell
 your
 first
 group
 they
 must
 give
 every
 group
 12
 pieces
 of
 candy.
 What
 is
 the
 
multiplication
 problem
 that
 would
 tell
 them
 how
 many
 pieces
 of
 candy
 they
 need?
 (12
 X
 5
 =
 60).
 Have
 
them
 write
 the
 problem
 on
 the
 board
 and
 explain
 to
 the
 class
 how
 they
 solved
 their
 problem.
 
 

When
 each
 group
 receives
 their
 candy
 from
 another
 group
 they
 should
 write
 down
 the
 problem
 
needed
 to
 show
 how
 many
 pieces
 of
 candy
 each
 student
 in
 the
 group
 will
 receive.
 (4
 Students
 X
 ?
 =
 
60).
 At
 the
 end
 of
 the
 lesson
 let
 the
 students
 eat
 their
 candy
 

Teacher
 lead
 class
 discussion
 
 

“Now
 we
 will
 discuss
 the
 section
 titled
 ‘What
 We
 Want
 to
 Know’.
 
 As
 we
 discuss
 the
 things
 we
 want
 to
 
learn
 about
 multiplication,
 you
 may
 write
 down
 your
 thoughts
 and
 ideas
 in
 the
 section
 titled
 ‘What
 I
 
Want
 to
 Know’.
 
 
 
 Who’s
 ready
 to
 begin?”
 

The
 Instructor
 will
 ask
 for
 volunteers
 to
 tell
 the
 class
 what
 they
 hope
 to
 find
 out
 by
 studying
 this
 unit.
 
Record
 the
 answers
 on
 the
 classroom
 KWL
 chart.
 
 
 

Journal
 Time
 

Students
 may
 record
 what
 they
 hope
 to
 learn
 in
 their
 journals.
 

Explanation
 (15
 minutes):
 Setting
 the
 agenda
 

The
 instructor
 will
 explain
 that
 we
 are
 going
 to
 be
 studying
 multiplication
 for
 the
 next
 unit:
 the
 agenda
 
for
 the
 unit:
 
 

• Students
 will
 bring
 home
 their
 journals
 daily
 and
 record
 their
 observations
 and
 discoveries
 
about
 multiplication
 in
 their
 journal
 

• Students
 will
 locate
 arrays
 in
 real
 life,
 and
 either
 photograph
 them,
 draw
 them,
 or
 bring
 in
 
examples
 of
 them.
 

• Students
 will
 create
 their
 own
 examples
 of
 multiplication
 through
 literature,
 music,
 and
 art.
 
• As
 we
 study
 certain
 aspects
 of
 multiplication,
 you
 will
 record
 your
 data
 and
 observations
 in
 

your
 journals.
 
• You
 will
 occasionally
 have
 other
 assignments
 that
 are
 to
 be
 recorded
 in
 your
 journals
 as
 well.
 
 I
 

will
 give
 you
 that
 information
 when
 we
 get
 to
 the
 appropriate
 lesson.
 

An
 instructor
 led
 Exploration
 of
 the
 journals:
 

• The
 instructor
 will
 show
 the
 students
 an
 example
 of
 the
 chart
 to
 record
 multiplication
 facts.
 
• The
 instructor
 will
 show
 the
 students
 an
 example
 of
 the
 charts
 and
 templates
 they
 will
 use
 

during
 this
 unit.
 
 The
 instructor
 will
 remind
 them
 that
 we
 will
 not
 begin
 the
 individual
 lessons
 
or
 activities
 until
 we
 have
 done
 them
 as
 a
 class.
 

7

Lesson
 2:
 
 Fish
 Bowl
 

Grade
 Level
  Introduction:
 
 This
 lesson
 introduces
 students
 to
 the
 concept
 of
 “multiplication
 as
 
repeated
 addition
 of
 equal
 sets.”
 First
 they
 will
 work
 either
 independently
 or
 in
 pairs
 to
 
Third
 and
 Fourth
  write
 and
 illustrate
 their
 solution
 to
 a
 “How
 many
 are
 there
 altogether?”
 problem,
 

  taking
 time
 to
 explore
 their
 thinking
 and
 clarify
 their
 understanding.
 Next,
 students
 will
 

  share
 their
 ideas
 with
 the
 class
 so
 others
 can
 try-­‐on
 alternate
 ways
 of
 visualizing
 
Time
 Needed
  solutions
 to
 the
 same
 problem.
 

 
 
 
50
 minutes
  Preparation:
 Prepare
 ahead
 of
 time
 small
 “packets”
 of
 Unifix
 cubes,
 one
 color
 for
 each
 

  work
 group,
 three
 cubes
 for
 each
 student.
 

 
 
Materials
  GLCE:
 3.N.MR.03.10
 Recognize
 situations
 that
 can
 be
 solved
 using
 multiplication
 and
 
division
 including
 finding
 "How
 many
 groups?"
 and
 "How
 many
 in
 a
 group?"
 and
 write
 
Clear
 container
  mathematical
 statements
 to
 represent
 those
 situations.
 
 
for
 “fishbowl”
 
Engagement
 (5
 minutes):
 First
 divide
 the
 class
 up
 into
 even
 groups
 of
 3-­‐5
 students
 
Unifix
 cube
 “fish”
  each.
 Then
 pass
 out
 small
 bins
 of
 Unifix
 cubes
 to
 each
 group,
 giving
 each
 a
 single,
 
(One
 color
 for
  unique
 color.
 Next,
 hold
 up
 a
 clear
 container
 (bowl,
 plastic
 bin,
 etc.).
 Tell
 the
 students
 
each
 work
 group,
  that
 it’s
 a
 “fishbowl”
 and
 you
 want
 each
 of
 them
 to
 put
 three
 “fish”
 from
 their
 group’s
 
three
 cubes
 for
  bin
 into
 the
 bowl.
 
each
 student)
 
 
Exploration
 (15
 minutes):
 After
 discussing
 how
 many
 students
 put
 fish
 into
 the
 bowl,
 
11”
 x
 14”
 paper
  tell
 the
 class
 that
 you
 want
 to
 see
 if
 they
 can
 figure
 out
 how
 many
 are
 in
 the
 bowl
 
altogether.
 On
 the
 board
 write:
 
Writing
 pencils
 

  There
 are
 ____
 fish
 in
 the
 bowl.
 
Colored
 pencils
 
  I
 think
 this
 because
 __________.
 
 

Grid
 paper
 
  Tell
 them
 they
 can
 work
 in
 pairs
 or
 independently,
 but
 they
 need
 to
 explain
 their
 
(For
 extension)
  thinking
 with
 numbers
 and
 words.
 They
 can
 use
 pictures
 too
 if
 that
 would
 help.
 

 

  Explanation
 (30
 minutes):
 Reconvene
 as
 a
 group
 and
 ask
 the
 students
 to
 share
 their
 
thinking
 with
 the
 class.
 Acknowledge
 the
 different
 responses
 by
 asking
 thoughtful
 
questions
 that
 extend
 their
 thinking
 and
 illuminate
 fuzzy
 logic.
 
 
Students
 might
 show
 some
 of
 the
 following
 examples
 (24
 students,
 6
 groups
 of
 4):
 

a)
 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1
 (etc)
 =
 72
 fish
 

b)
 3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3
 =
 72
 fish
 

c)
 
 3+3+3+3
 =
 12
 red
 fish
  3+3+3+3
 =
 12
 blue
 fish
  3+3+3+3
 =
 12
 green
 fish
 


 
 
 
 
 3+3+3+3
 =
 12
 yellow
 fish
  3+3+3+3
 =
 12
 brown
  3+3+3+3
 =
 12
 orange
 fis

   


  Then
 add
 12+12+12+12+12+12
 and
 you
 get
 72
 fish!
 

d)
 12
 red
 fish
 +
 12
 yellow
 +
 12
 blue
 +
 12
 brown
 +
 12
 green
 +
 12
 orange
 =
 72
 fish
 

e)
 6
 groups
 of
 kids
 x
 4
 kids
 in
 each
 group
 x
 3
 fish
 for
 each
 kid
 =
 72
 fish
 

f)
 
 24
 kids
 x
 3
 fish
 each
 =
 72
 fish
 

8

 
This
 is
 the
 time
 to
 explicitly
 make
 the
 connection
 that:
 


 

1) You
 can
 make
 groups
 of
 like
 things
 and
 add
 them
 together.
 
 

“What
 kinds
 of
 groups
 do
 we
 see
 here?
 72
 fish
 1
 time.
 3
 fish
 24
 times.
 6
 groups
 of
 three
 fish
 added
 
together.
 12
 groups
 of
 fish
 6
 times.”
 
 

2) Adding
 up
 groups
 of
 things
 is
 quicker
 and
 easier
 than
 adding
 up
 singletons
 (a
 and
 b).
 

“For
 those
 of
 you
 who
 added
 each
 fish
 by
 itself
 up
 to
 72
 and
 those
 who
 added
 3
 fish
 24
 times,
 did
 
you
 have
 any
 problems
 with
 your
 strategy?
 Do
 any
 other
 strategies
 look
 easier
 or
 faster?”
 

3) Multiplication
 is
 repeated
 addition
 of
 similar
 sized
 groups
 of
 things
 (c
 and
 d).
 
“Who
 can
 explain
 what
 I
 mean
 by,
 ‘multiplication
 is
 repeated
 addition’?”
 

4) Multiplication
 is
 commutative
 like
 addition,
 that
 is
 2x4
 =
 4x2
 =
 8.
 

On
 the
 board
 write:
 
  12
 =
 3
 x
 4
 =
 4
 x
 3
 =
 12
 

 
 
 
 

“Who
 can
 explain
 what
 I
 mean
 by,
 ‘multiplication
 is
 commutative’?”
 

5) You
 can
 group
 like
 numbers
 of
 things
 and
 add
 them
 to
 groups
 with
 larger
 or
 smaller
 numbers.
 
 

“Can
 you
 figure
 out
 how
 many
 fish
 there
 would
 be
 if
 there
 were
 5
 students
 in
 the
 green
 and
 orange
 
fish
 groups?”
 


  Sample
 answer:
 


  3+3+3+3
 =
 12
 red
 fish
 
  3+3+3+3
 =
 12
 blue
 fish
 
  3+3+3+3+3
 =
 15
 green
 fish
 

 
 
 
 
 
 
 3+3+3+3
 =
 12
 yellow
 fish
  3+3+3+3
 =
 12
 brown
 fish
  3+3+3+3+3
 =
 15
 orange
 fish
 


 
 
  12
 x
 4
 =
 48
 fish
 and
 15
 x
 2
 =
 30
 fish
 


 
 
  48
 fish
 +
 30
 fish
 =
 78
 fish!
 

“What
 if
 there
 were
 8
 students
 had
 red
 and
 8
 had
 yellow
 fish,
 5
 had
 blue
 and
 5
 had
 brown
 fish
 and
 
only
 3
 students
 had
 green
 and
 3
 had
 orange?”
 


  Sample
 answer:
 


  3+3+3+3
 =
 12
 red
 fish
 
  3+3+3+3+3
 =
 15
 blue
 fish
  3+3+3
 =
 9
 green
 fish
 

 
 
 
 
 
 
 3+3+3+3
 =
 12
 yellow
 fish
  3+3+3+3+3
 =
 15
 brown
 fish
  3+3+3
 =
 9
 orange
 fish
 


 
  12
 x
 2
 =
 24
 fish
 and
 15
 x
 2
 =
 30
 fish
 and
 9
 x
 2
 =
 18
 fish
 

 
  24
 fish
 +
 30
 fish
 +
 18
 fish
 =
 72
 fish!
 
 

“Hey,
 that’s
 interesting.
 That’s
 the
 same
 amount
 as
 we
 had
 the
 first
 time!
 Who
 knows
 why?”
 

6) Multiplying
 groups
 of
 things
 is
 even
 quicker
 and
 easier
 when
 you
 learn
 your
 math
 facts!
 


 

3
 red
 x
 4
 kids
 =
 12
 red
 fish
 
 
 
 
 3
 blue
 x
 4
 kids
 =
 12
 blue
 fish
 
  3
 green
 x
 4
 kids
 =
 12
 green
 fish
 
 
3
 yellow
 x
 4
 kids
 =
 12
 yellow
 fish
 
 
 3
 brown
 x
 4
 kids
 =
 12
 brown
 fish
 
 
 3
 orange
 x
 4
 kids
 =
 12
 orange
 fish
 


 
 
  And
 12
 fish
 x
 6
 groups
 =
 72
 fish!
 

9

“There’s
 one
 way
 that’s
 even
 faster.
 Can
 anyone
 see
 it?
 …
 3
 fish
 x
 24
 kids
 =
 72
 fish!”
 
“Do
 you
 think
 you
 could
 make
 similar
 groups
 for
 the
 other
 examples?
 Sure
 you
 could.
 
Who
 wants
 to
 show
 us
 how?”
 
Invite
 at
 least
 two
 students
 come
 to
 the
 board
 and
 show
 their
 thinking.
 Ask
 the
 rest
 of
 the
 
class
 if
 they
 agree.
 Be
 sure
 to
 ask
 them
 to
 explain
 their
 thinking
 if
 they
 head
 down
 the
 
wrong
 path.
 Others
 in
 the
 class
 who
 may
 have
 gone
 there
 too
 will
 benefit.
 

 
Extension:
 Pass
 out
 grid
 paper
 and
 ask
 the
 students
 to
 represent
 their
 thinking
 in
 colorful
 arrays.
 Ask
 
them
 to
 write
 a
 number
 sentence
 that
 means
 the
 same
 thing
 as
 their
 array.
 Ask
 for
 volunteers
 to
 
explain
 their
 work.
 Ask
 thoughtful
 questions
 that
 extend
 their
 thinking
 and
 illuminate
 fuzzy
 logic.
 

 
Evaluation:
 
 Monitor
 student’s
 oral
 and
 written
 responses
 to
 assess
 understanding
 of
 multiplication
 as
 
repeated
 addition.
 Collect
 written
 responses
 as
 formative
 assessment.
 

 

 
Reference:
 Burns,
 M.
 (1995).
 Writing
 in
 Math
 Class:
 A
 Resource
 for
 Grades
 2-­‐8.
 Sausalito,
 CA:
 Math
 
Solutions.
 

10

Lesson
 3:
 
 Groupings
 All
 Around
 Us
 

Grade
 Level
  Introduction:
 
 This
 lesson
 introduces
 students
 to
 the
 concept
 of
 “Multiplication
 is
 a
 
quick
 way
 to
 figure
 out
 how
 many
 you
 have
 altogether
 of
 something
 when
 things
 come
 
Third
 and
 Fourth
  in
 groups.”
 First
 the
 class
 will
 work
 collaboratively
 brainstorming
 a
 list
 of
 objects
 in
 the
 

  world
 that
 always
 occur
 in
 groups
 of
 2,
 3,
 4
 …
 12
 and
 solving
 made
 up
 problems
 to
 find
 

  “how
 many?”.
 Next,
 students
 will
 work
 in
 small
 groups
 creating
 and
 solving
 their
 own
 
Time
 Needed
  made
 up
 problems.
 Finally,
 each
 group
 will
 share
 their
 ideas
 with
 the
 class
 so
 others
 
can
 try-­‐on
 alternate
 ways
 of
 visualizing
 solutions
 to
 multiplication
 problems.
 
50
 minutes
 
 
 
 

  Preparation:
 Prior
 to
 beginning
 the
 lesson,
 determine
 how
 the
 class
 will
 be
 divided
 up
 

  into
 groups
 of
 3-­‐5
 students
 each.
 Have
 sufficient
 newsprint
 for
 each
 group
 to
 have
 one
 
Materials
  piece.
 Also,
 be
 prepared
 to
 record
 lists
 generated
 by
 the
 class
 as
 a
 whole
 on
 newsprint
 
posted
 on
 the
 wall,
 chart
 paper
 on
 an
 easel
 or
 on
 the
 white
 board.
 For
 the
 extension
 
Newsprint
 (One
  activity,
 each
 student
 will
 also
 need
 a
 piece
 of
 paper
 on
 which
 to
 write
 and
 illustrate
 a
 
piece
 for
 each
  sample
 multiplication
 problem.
 
work
 group)
 
 
GLCE:
 3.N.MR.03.10
 Recognize
 situations
 that
 can
 be
 solved
 using
 multiplication
 and
 
Drawing
 paper
  division
 including
 finding
 "How
 many
 groups?"
 and
 "How
 many
 in
 a
 group?"
 and
 write
 
(At
 least
 one
 for
  mathematical
 statements
 to
 represent
 those
 situations.
 
 
each
 student)
 
3.N.FL.03.11
 Find
 products
 fluently
 up
 to
 10
 x
 10;
 find
 related
 quotients
 using
 
11”
 x
 14”
 paper
  multiplication
 and
 division
 relationships.
 
 

Writing
 pencils
  Engagement
 (10
 minutes):
 “Today
 we
 are
 going
 to
 brainstorm
 what
 sorts
 of
 things
 that
 
come
 in
 groups
 of
 2s,
 3s,
 4s,
 5s
 all
 the
 way
 up
 to
 12s.
 First
 we’re
 going
 to
 list
 as
 a
 class
 
Colored
 pencils
  together
 examples
 of
 things
 that
 come
 in
 groups
 of
 two.
 Then,
 we’re
 going
 to
 break
 out
 

  into
 small
 groups.
 Each
 group
 will
 continue
 to
 brainstorm
 lists
 of
 things
 that
 come
 in
 
groups
 of
 3s
 through
 12s
 and
 record
 their
 ideas
 on
 a
 large
 sheet
 of
 paper.”
 

 
 
Now,
 together
 as
 a
 class,
 brainstorm
 a
 list
 of
 things
 that
 always
 come
 in
 twos,
 excluding
 
things
 that
 sometimes
 come
 in
 twos.
 If
 students
 are
 unsure
 about
 an
 item,
 list
 it
 off
 to
 
the
 side
 to
 research
 later.
 Once
 you
 have
 a
 good
 list
 of
 items,
 break
 up
 into
 the
 smaller
 
work
 groups
 for
 the
 students
 to
 continue
 on
 their
 own.
 Be
 sure
 to
 remind
 them
 that
 
since
 they
 are
 not
 listing
 groups
 of
 1s
 and
 you
 have
 already
 listed
 groups
 of
 2s
 
together,
 each
 group
 will
 be
 exploring
 10
 lists
 total.
 

 
Exploration
 (20
 minutes):
 The
 first
 challenge
 of
 this
 activity
 will
 arise
 as
 the
 students
 
figure
 out
 how
 they
 will
 work
 cooperatively
 to
 brainstorm
 and
 record
 their
 groups’
 
lists.
 Resist
 the
 urge
 to
 step
 in,
 confidently
 assuring
 them
 that
 they
 can
 figure
 it
 out
 for
 
themselves.
 The
 next
 puzzle
 will
 be
 to
 figure
 out
 how
 to
 arrange
 their
 thinking
 on
 the
 
large
 sheet
 of
 paper.
 Again,
 resist
 the
 urge
 to
 step
 in.
 Use
 this
 time
 to
 assess
 the
 
creativity
 and
 uniqueness
 of
 each
 student’s
 thinking,
 as
 well
 as
 the
 students’
 ability
 to
 
cooperatively
 problem
 solve
 in
 a
 group
 setting.
 
 

 
Explanation
 (20
 minutes):
 Once
 all
 the
 groups
 have
 completed
 their
 lists,
 it’s
 time
 to
 
discuss
 them
 together
 as
 a
 class.
 

 

11

“Now
 we’ll
 go
 around
 the
 room,
 group
 by
 group.
 Each
 group
 will
 report
 just
 one
 thing
 from
 any
 one
 list,
 
without
 telling
 us
 which
 list
 it’s
 on.
 Then
 the
 others
 in
 the
 class
 will
 have
 the
 chance
 to
 decide
 where
 it
 
belongs.
 Once
 we
 agree,
 I’ll
 write
 it
 on
 the
 board
 under
 the
 correct
 number.
 Since
 you’ll
 want
 to
 report
 
something
 from
 your
 list
 that
 has
 not
 already
 been
 suggested,
 take
 a
 few
 minutes
 now
 to
 have
 an
 
alternative
 in
 case
 the
 one
 you
 chose
 has
 already
 been
 mentioned.”
 

 
This
 part
 of
 the
 activity
 will
 involve
 group
 thinking
 and
 discernment.
 Some
 items
 will
 be
 obvious,
 legs
 
on
 a
 dog
 and
 cans
 in
 a
 six-­‐pack,
 for
 example;
 others
 may
 not
 be,
 such
 as
 legs
 on
 a
 stool
 or
 points
 on
 a
 
star.
 You
 will
 need
 to
 talk
 this
 through
 problem
 together.
 Someone
 may
 suggest
 something
 that
 makes
 
no
 sense.
 Others
 may
 be
 very
 creative,
 so
 be
 sure
 to
 ask
 students
 to
 explain
 their
 thinking.
 For
 
example,
 a
 group
 my
 say
 “four
 holes
 in
 a
 shirt,”
 then
 offer
 they
 were
 thinking
 of
 the
 one
 for
 the
 neck,
 
at
 the
 bottom
 and
 for
 each
 sleeve!
 
 

 
Extension:
 These
 lists
 are
 a
 rich
 resource
 for
 generating
 problems
 that
 students
 can
 solve.
 Start
 by
 
creating
 problems
 and
 linking
 them
 to
 their
 proper
 multiplication
 sentences.
 
 

1) For
 example,
 ask:
 “How
 many
 cans
 of
 Coke
 are
 in
 three
 six
 packs?”
 
 

If
 the
 students
 are
 able,
 have
 them
 tell
 you
 what
 sentence
 to
 write.
 If
 not,
 you
 write
 3
 x
 6
 =
 18
 on
 
the
 board.
 Then
 ask:
 “What
 does
 the
 6
 tell
 us?
 What
 does
 the
 3
 tell
 us?
 What
 does
 the
 18
 tell
 us?
 
How
 do
 you
 know
 that
 18
 is
 correct?”
 

2) Another
 activity
 would
 be
 to
 have
 students
 to
 write
 and
 illustrate
 multiplication
 problems
 for
 
others
 to
 solve.
 They
 can
 write
 the
 problem
 out
 in
 words
 with
 an
 illustration
 on
 one
 side
 of
 the
 
paper,
 then
 turn
 it
 over
 and
 write
 the
 complete
 multiplication
 sentence
 on
 the
 other
 side.
 That
 
way,
 children
 can
 read
 each
 other’s
 problem,
 solve
 them,
 and
 check
 their
 solutions.
 Challenge
 
students
 to
 see
 how
 many
 ways
 they
 can
 figure
 out
 the
 answer.
 Then
 ask
 volunteers
 to
 share
 their
 
multiplication
 problems
 and
 their
 thinking
 for
 how
 they
 solved
 them.
 

It’s
 important
 that
 the
 solution
 is
 more
 than
 the
 answer
 that
 results
 from
 the
 multiplication;
 it
 is
 the
 
entire
 multiplication
 sentence.
 The
 emphasis
 is
 on
 relating
 the
 multiplication
 sentence
 to
 the
 problem
 
situation
 to
 develop
 children’s
 understanding.
 

3)
  Another
 extension
 activity
 would
 be
 to
 generate
 charts
 from
 the
 lists
 of
 12
 multiples.
 For
 example:
 

 

People
  Eyes
  Multiplication
 Sentence
 

1
  2
  1
 x
 2
 =
 2
 

2
  4
  2
 x
 2
 =
 4
 

3
  6
  3
 x
 2
 =
 6
 

(etc.)
  (etc.)
  (etc.)
 


 
Evaluation:
 
 Monitor
 student’s
 oral
 and
 written
 responses
 to
 assess
 understanding
 of
 multiplication
 as
 a
 
quick
 way
 to
 figure
 out
 how
 many
 you
 have
 altogether
 of
 something
 when
 things
 come
 in
 groups,
 as
 
well
 as
 their
 ability
 to
 work
 in
 groups
 effectively
 together.
 Collect
 written
 responses
 as
 formative
 
assessment.
 Use
 extensions
 to
 challenge
 students
 who
 already
 have
 a
 basic
 understanding
 of
 
multiplication
 or
 to
 provide
 additional
 practice
 to
 students
 who
 need
 help
 clarifying
 their
 understanding.
 

 

 
Reference:
 Burns,
 M.
 (1987).
 A
 Collection
 of
 Math
 Lessons:
 From
 Grades
 3
 through
 6.
 Sausalito,
 CA:
 Math
 
Solutions.
 

12

 

Lesson
 4:
 Circles
 and
 Stars
 Game
 

Grade
 Level
  Introduction:
 Through
 this
 game,
 students
 learn
 to
 see
 multiplication
 as
 the
 combining
 
of
 equal-­‐size
 groups
 that
 can
 be
 represented
 with
 a
 multiplication
 equation.
 
Third
 and
 Fourth
 
 

  Preparation:
 Divide
 the
 class
 up
 into
 groups
 of
 two
 to
 four
 students
 and
 distribute
 

  materials
 accordingly.
 
 
Time
 Needed
 
 
GLCE:
 3.N.FL.03.11
 Find
 products
 fluently
 up
 to
 10
 x
 10;
 find
 related
 quotients
 using
 
50
 minutes
  multiplication
 and
 division
 relationships.
 
 

 

  Engagement
 (10
 minutes):
 Invite
 the
 children
 to
 fold
 their
 pieces
 of
 paper
 in
 half,
 then
 
Materials
  in
 half
 again,
 creating
 four
 quadrants
 on
 each
 side.
 Explain
 the
 rules
 of
 the
 game.
 

One
 six-­‐sided
 die
  1.
 The
 first
 player
 starts
 the
 first
 round
 by
 rolling
 the
 die.
 This
 number
 is
 the
 amount
 of
 
(One
 die
 per
  circles
 he/she
 will
 draw
 in
 the
 first
 square
 on
 his/her
 paper.
 It
 is
 also
 the
 first
 
group
 of
 2-­‐4
  number
 in
 his/her
 multiplication
 problem.
 
 
students)
 
2.
 The
 player
 rolls
 the
 die
 again.
 This
 number
 is
 the
 amount
 of
 stars
 he/she
 will
 draw
 in
 
Three
 8
 ½”
 x
 11”
  each
 circle
 in
 that
 first
 square.
 It
 is
 also
 the
 second
 number
 in
 his/her
 multiplication
 
sheet
 of
 paper
  problem.
 
 
for
 each
 student
 
3.
 
 Now
 the
 player
 writes
 the
 two
 numbers
 and
 the
 answer
 in
 a
 multiplication
 sentence
 
Writing
 pencils
  right
 below
 the
 circles
 and
 stars.
 

12-­‐sided
 dice
 
 
  4.
 Each
 player
 takes
 a
 turn
 until
 the
 group
 has
 repeated
 filled
 in
 all
 eight
 squares
 on
 
(For
 extension)
  their
 score
 sheets
 (front
 and
 back).
 


  5.
 
 Add
 up
 all
 of
 your
 answers.
 Whoever
 has
 the
 most
 wins
 the
 game!
 

Model
 how
 to
 play
 the
 game
 then
 invite
 the
 class
 to
 play
 one
 round
 with
 guided
 
practice.
 

 
Exploration
 (15
 minutes):
 The
 students
 play
 several
 rounds
 of
 Circles
 and
 Stars.
 
 

 
Explanation
 (15
 minutes):
 Pose
 the
 following
 questions
 for
 students
 to
 discuss
 in
 small
 
groups
 or
 as
 a
 class.
 

-­‐
 What
 is
 the
 fewest
 number
 of
 stars
 you
 can
 get
 in
 one
 round?
 Explain.
 

-­‐
 What
 is
 the
 greater
 number
 of
 stars
 you
 can
 get
 in
 one
 round?
 Explain.
 

-­‐
 What
 other
 observations
 did
 you
 make
 as
 you
 were
 playing
 this
 game?
 Explain.
 

-­‐
 What
 numbers
 did
 you
 represent
 in
 different
 ways?
 Compare
 with
 your
 partner.
 
Explain.
 

-­‐
 I
 have
 a
 die
 that
 has
 a
 0.
 What
 would
 you
 do
 if
 your
 first
 roll
 was
 a
 zero?
 Explain.
 

-­‐
 What
 would
 you
 do
 if
 your
 first
 roll
 was
 a
 5
 and
 your
 second
 roll
 was
 a
 zero?
 
Explain.
 

Create
 Class
 Data
 Chart.
 (Prepare
 before
 the
 lesson.)
 List
 all
 numbers
 1-­‐36
 on
 a
 chart
 

13

using
 column
 format.
 (Thirty-­‐six
 is
 the
 largest
 product
 possible
 using
 a
 six-­‐sided
 die.)
 
 
Select
 one
 student
 to
 bring
 up
 one
 of
 his/her
 recording
 sheet.
 Together
 model
 how
 to
 use
 tally
 marks
 
to
 record
 the
 student’s
 scores
 for
 each
 round
 on
 the
 Class
 Data
 Chart.
 Then
 invite
 the
 groups
 to
 come
 
up
 and
 record
 their
 scores
 from
 all
 of
 their
 games
 on
 the
 Class
 Data
 Chart.
 Suggest
 that
 if
 one
 partner
 
reads
 each
 score,
 the
 other
 partner
 can
 record
 tally
 marks.
 
 
Discuss
 the
 data.
 After
 all
 students
 have
 played
 several
 games
 and
 recorded
 their
 products
 for
 each
 
round
 on
 the
 class
 chart,
 engage
 students
 in
 conversation
 about
 the
 data
 chart,
 asking
 questions
 like:
 
 

-­‐
 
 Why
 did
 I
 write
 the
 numbers
 1-­‐36
 on
 the
 chart?
 
-­‐
 
 Are
 there
 numbers
 that
 are
 impossible
 using
 a
 1-­‐6
 die?
 Explain.
 
-­‐
 
 Why
 do
 some
 numbers
 have
 more
 tally
 marks
 than
 other
 numbers?
 Explain.
 
-­‐
 
 What
 are
 the
 ways
 to
 get
 2
 as
 an
 answer?
 Ways
 for
 6?
 Ways
 for
 12?
 (Students
 might
 think
 about
 

this
 with
 a
 partner
 or
 in
 small
 groups.
 Record
 equations.)
 
-­‐
 
 Which
 number(s)
 1-­‐36
 has
 the
 most
 combinations
 using
 two
 1-­‐6
 dice?
 What
 numbers
 can
 I
 skip
 

count
 by
 to
 say
 this
 number?
 (Relate
 numbers
 on
 dice
 to
 factors
 in
 multiplication
 equations.
 
-­‐
 
 You
 can
 skip
 count
 by
 both
 factors
 to
 figure
 out
 the
 product.
 Is
 this
 always
 true?
 (Ask
 students
 to
 

test
 this
 idea.
 Some
 may
 want
 to
 test
 larger
 numbers.)
 
-­‐
 
 Is
 there
 a
 product
 that
 can
 only
 be
 represented
 one
 way?
 Why?
 Explain.
 
-­‐
 
 What
 other
 observations
 do
 you
 notice
 about
 the
 data?
 
 
-­‐
 
 How
 might
 this
 data
 be
 useful
 for
 thinking
 about
 multiplication
 combinations
 (facts)?
 

 

Extension:
 Invite
 those
 looking
 for
 a
 challenge
 to:
 

1.
 Change
 the
 die
 to
 a
 higher
 number
 sided
 die
 (e.g.
 12
 sided)
 to
 make
 the
 multiplication
 problems
 
more
 difficult.
 

2.
 Use
 two
 dice
 at
 the
 same
 time
 and
 choose
 which
 order
 to
 put
 them
 in
 for
 your
 circles
 and
 stars.
 
Commutative
 property
 of
 multiplication
 rule
 says
 you
 get
 the
 same
 answer
 no
 matter
 what
 order.
 

3.
 Write
 the
 fact
 family
 for
 each
 problem
 you
 roll
 to
 practice
 multiplication
 and
 division
 sentences.
 

 
  Example:
 
 3
 x
 4
 =
 12
 
 
 4
 x
 3
 =
 12
 
 
 
 12
 ÷
 3
 =
 4
 
 
 
 12
 ÷
 4
 =
 3
 

 
Evaluation:
 
 Monitor
 student’s
 oral
 and
 written
 responses
 to
 assess
 understanding
 of
 multiplication
 as
 
repeated
 addition.
 Collect
 score
 sheets
 for
 formative
 assessment.
 

 

 
Reference:
 Cleveland
 County
 Schools.
 http://tinyurl.com/circlesandstarsdirections.
 Accessed
 November
 26,
 
2010.
 (Adapted
 from
 Math
 by
 All
 Means;
 Multiplication
 Grade
 3
 by
 Marilyn
 Burns.)
 

14

Circles and Stars Multiplication Game

Mary Ebejer and Becki West

Objective
Students will be able to form simple multiplication problems using 1 die by grouping them
in circles and using stars to represent the numbers then multiplying them to find the
product.

Materials Needed **
1 Die (6 sided)
Paper 2
 
 
 x
 
 
 4
 
 
 =
 
 
 8
 
Pencil

Directions

1. Fold the paper into separate sections, usually four squares on front and four on back.

2. The first player starts the first round by rolling the die. This number is the amount of
circles he/she will draw in the first square on his/her score sheet. It is also the first

number in his/her multiplication problem.

3. The first player rolls the die again. This number is the amount of stars he/she will draw
in each circle in that first square. It is also the second number in his/her multiplication
problem.

4. Now the player writes the two numbers and the answer in a multiplication sentence
right below the circles and stars.

5. Each player takes a turn until the group has filled in all eight squares on their score
sheets (front and back).

6. Add up all of your answers. Whoever has the most wins the game!

Challenges

1. Change the die to a higher number sided die (e.g. 12 sided) to make the multiplication
problems more difficult.

2. Use two dice at the same time and choose which order to put them in for your circles
and stars. Commutative property of multiplication rule says you get the same answer
no matter what order.

3. Write the fact family for each problem you roll to practice multiplication and division
sentences.

Example: 3 x 4 = 12 4 x 3 = 12 12 ÷ 3 = 4 12 ÷ 4 = 3

From: http://tinyurl.com/circlesandstarsgame. Accessed November 26, 2010. (Variation on Marilyn Burn:
“Circles and Stars.” Math By All Means. ©1991 The Math Solution Publications.)

15

Lesson
 5:
 
 Creating
 Multiplication
 Tables
 

Grade
 Level
  Introduction:
 
 In
 this
 5-­‐day
 lesson,
 students
 will
 create
 arrays
 for
 multiplication
 
Third
 and
 Fourth
  fact
 families
 0-­‐12
 and
 cleverly
 transfer
 them
 to
 create
 a
 multiplication
 table
 to
 

  laminate
 for
 their
 own
 personal
 use.
 

 
 
Time
 Needed
  Preparation:
 Prior
 to
 beginning
 the
 lesson,
 ask
 students
 to
 respond
 to
 this
 prompt
 in
 
5
 days
  their
 math
 journals:
 
50
 minutes/day
 

 
  Write
 what
 you
 know
 about
 the
 0-­‐12
 multiplication
 table.
 

 
Materials
  Their
 response
 will
 serve
 as
 a
 benchmark
 for
 their
 formative
 assessments.
 
 
For
 each
 group:
 
 
24
 1”
 square
 tiles
  GLCE:
 3.N.FL.03.11
 Find
 products
 fluently
 up
 to
 10
 x
 10;
 find
 related
 quotients
 using
 
multiplication
 and
 division
 relationships.
 
 
For
 each
 student:
 
 
8
 ½”
 x
 11”
 paper
  DAY
 ONE:
 MAKING
 RECTANGLES
 
ruled
 with
 ½”
 
 
squares
 (stack
 of
  Engagement
 (10
 minutes):
 Divide
 the
 class
 up
 into
 groups
 of
 four
 students.
 Invite
 
extras
 on
 hand)
  one
 person
 from
 each
 group
 to
 come
 up
 and
 count
 out
 25
 tiles
 and
 bring
 them
 back
 
to
 their
 group.
 
Writing
 pencils
 
 
Exploration
 (40
 minutes):
 “Each
 group
 will
 have
 25
 tiles.
 I
 would
 like
 you
 to
 work
 with
 
Colored
 pencils
  a
 partner
 in
 your
 group
 for
 this
 first
 task.
 (A
 group
 of
 three
 will
 work
 if
 there
 is
 an
 odd
 
number.)
 I
 want
 you
 and
 your
 partner
 to
 take
 12
 tiles
 and
 arrange
 them
 into
 a
 solid
 
Scissors
  rectangle.
 Your
 rectangle
 should
 be
 all
 filled
 in
 completely.
 Don’t
 use
 the
 tiles
 just
 to
 
outline
 a
 rectangle.”
 
“Rectangles”
 
Worksheet
  Students
 create
 their
 rectangles.
 

 
“Look
 at
 your
 group’s
 rectangles.
 Raise
 your
 hand
 if
 both
 the
 rectangles
 are
 the
 same.”
 

“Now
 raise
 your
 hand
 if
 your
 rectangles
 are
 different.”
 

Some
 may
 not
 raise
 their
 hands
 at
 all
 because
 they
 have
 the
 same
 shape
 but
 a
 
different
 orientation,
 e.g.
 6x2
 and
 2x6
 or
 4x3
 and
 3x4.
 Ask
 the
 students
 to
 describe
 
their
 rectangles
 so
 you
 can
 draw
 them
 on
 the
 board.
 Show
 that
 the
 rectangles
 are
 
the
 same
 dimension,
 just
 in
 a
 different
 position,
 so
 they
 are
 the
 same.
 

Rectangles
 that
 are
 the
 same
 shape
 and
 orientation
 but
 used
 different
 colors
 are
 also
 
the
 same.
 

Have
 a
 member
 from
 each
 group
 come
 up
 and
 draw
 their
 rectangle
 on
 the
 board
 
until
 all
 factors
 of
 12
 are
 represented
 (1x12;
 2x6;
 3x4).
 Ask
 them
 to
 write
 “12”
 on
 
each
 rectangle.
 

“Let’s
 try
 another
 number.
 This
 time,
 work
 as
 a
 group
 instead
 of
 with
 a
 partner.
 See
 if
 
you
 can
 find
 all
 the
 ways
 to
 build
 rectangles
 with
 sixteen
 tiles.
 Draw
 each
 rectangle
 
you
 find
 on
 the
 grid
 paper,
 write
 16
 inside,
 and
 cut
 it
 out.
 If
 you
 finish
 that
 and
 others
 
are
 still
 working,
 do
 the
 same
 for
 the
 number
 7.”
 (Write
 16
 and
 7
 on
 the
 board.)
 

If
 anyone
 asks,
 a
 4x4
 square
 counts
 because
 a
 square
 is
 a
 rectangle.
 

Once
 you’re
 sure
 everyone
 understands
 the
 directions,
 they
 can
 continue
 making
 
rectangles
 for
 numbers
 1-­‐25.
 Suggest
 that
 they
 continue
 using
 the
 tiles
 if
 that
 helps.
 
 

16

“Draw
 each
 rectangle
 you
 find
 on
 the
 grid
 paper,
 write
 the
 number
 on
 it
 and
 cut
 it
 out.
 You
 will
 be
 
cutting
 out
 a
 lot
 of
 rectangles
 so
 draw
 them
 close
 together
 to
 conserve
 paper.
 Also,
 don’t
 forget
 
the
 number
 12.
 We
 already
 did
 it
 on
 the
 board,
 but
 you
 will
 need
 to
 draw
 and
 cut
 out
 rectangles
 for
 
that
 one
 too.
 Also,
 you
 will
 want
 to
 figure
 out
 a
 way
 to
 keep
 track
 of
 which
 ones
 you
 have
 finished.
 
So
 take
 a
 minute
 to
 get
 organized
 before
 you
 begin.
 Any
 questions?”
 

(If
 the
 paper
 isn’t
 long
 enough
 to
 cut
 out
 the
 longest
 rectangles,
 it’s
 okay
 to
 tape
 two
 pieces
 
together.)
 

As
 the
 time
 for
 the
 activity
 runs
 out,
 give
 each
 group
 a
 legal-­‐size
 envelope.
 Ask
 them
 to
 put
 their
 
names
 on
 it
 and
 put
 all
 of
 their
 rectangles
 inside,
 as
 well
 as
 any
 extra
 paper
 and
 scraps
 of
 paper
 
still
 big
 enough
 for
 more
 rectangles.
 Put
 their
 envelopes
 and
 tiles
 on
 the
 supply
 table.
 Tomorrow,
 
when
 it’s
 time
 for
 math,
 they
 can
 get
 their
 envelopes,
 some
 tiles
 and
 paper
 and
 continue
 working.
 


 
DAY
 TWO:
 FINISH
 RECTANGLES;
 BEGIN
 SUMMARIZING
 

 
Engagement
 (10
 minutes):
 On
 the
 board
 write
 the
 numbers
 1-­‐12
 across
 the
 top,
 with
 about
 6-­‐8”
 
between
 each.
 As
 the
 groups
 finish,
 ask
 them
 to
 organize
 their
 rectangles
 by
 number.
 Then
 ask
 
one
 group
 at
 a
 time
 to
 come
 tape
 their
 rectangles
 to
 the
 board
 under
 the
 corresponding
 number.
 
Be
 sure
 to
 ask
 if
 any
 other
 group
 has
 any
 other
 rectangles
 after
 each
 set
 of
 rectangles
 is
 posted.
 If
 
a
 group
 is
 missing
 a
 set
 or
 two
 of
 rectangles,
 this
 would
 be
 a
 good
 time
 to
 make
 them.
 

 
Explanation
 (40
 minutes):
 Distribute
 “Exploring
 Our
 Rectangles”
 worksheet
 to
 each
 student
 and
 
invite
 groups
 to
 investigate
 the
 patterns
 together.
 

You
 can
 leave
 the
 rectangles
 posted
 on
 the
 board
 for
 the
 next
 day’s
 lesson.
 


 
DAY
 THREE:
 MAKING
 OUR
 MULTIPLICATION
 TABLES
 

 
Engagement
 (10
 minutes):
 Invite
 the
 students
 to
 come
 up
 to
 the
 board
 to
 take
 a
 good
 look
 at
 al
 of
 
the
 rectangles
 they
 have
 posted.
 After
 a
 few
 minutes,
 invite
 them
 to
 sit
 down
 on
 the
 floor
 near
 the
 
rectangle
 display
 and
 ask
 them
 how
 it
 went
 working
 in
 groups
 on
 their
 rectangles.
 (“What
 worked
 
well?”
 “What
 could
 have
 gone
 better?”)
 
 

 
Exploration
 (40
 minutes):
 Work
 through
 each
 of
 the
 questions
 on
 the
 “Exploring
 Our
 Rectangles”
 
worksheet,
 listing
 the
 answers
 on
 the
 board,
 discussing
 the
 patterns,
 and
 giving
 new
 vocabulary
 
when
 appropriate.
 For
 example,
 for
 rectangles
 that
 have
 a
 side
 with
 two
 squares
 on
 them,
 write
 2,
 
4,
 6,
 8
 10.
 12,
 14,
 16,
 18,
 20,
 22,
 24.
 

“What
 do
 you
 notice
 about
 these
 numbers?”
 (They
 skip
 every
 other
 one.)
 

“Who
 could
 continue
 the
 numbers
 in
 this
 pattern?”
 

“What
 is
 another
 name
 for
 these
 numbers?”
 (Even)
 

“These
 numbers
 are
 also
 multiples
 of
 2
 because
 each
 can
 be
 written
 as
 two
 times
 something
 …
 2
 
times
 2
 is
 4
 (write
 2
 x
 2
 =
 4).”
 
 

Other
 patterns
 to
 make
 note
 of
 include
 multiples
 of
 3,
 4
 and
 5,
 as
 well
 as
 squares,
 like
 1,
 4,
 9,
 16
 
and
 25.
 Ones
 with
 only
 one
 rectangle
 like
 1,
 2,
 3,
 5,
 7,
 11,
 13,
 17,
 19,
 24
 are
 prime.
 

Next,
 introduce
 the
 idea
 of
 transferring
 their
 rectangles
 to
 a
 chart.
 

17

“Here’s
 what
 I
 want
 you
 to
 do
 next.
 I’ll
 demonstrate
 on
 the
 board;
 then
 you’ll
 each
 do
 this
 
individually.
 You’ll
 use
 your
 own
 sheet
 of
 squared
 paper,
 but
 you’ll
 share
 your
 group’s
 rectangles.”
 

Tape
 a
 piece
 of
 the
 squared
 paper
 to
 the
 board.
 Take
 the
 3-­‐by-­‐4
 rectangle
 and
 place
 it
 on
 the
 
squared
 paper
 in
 the
 upper
 left-­‐hand
 corner.
 Then
 lift
 the
 lower
 right-­‐hand
 corner
 and
 write
 the
 
number
 12
 in
 the
 square.
 Explain:
 

“If
 I
 drew
 a
 rectangle
 around
 the
 12,
 I
 would
 outline
 the
 3-­‐by-­‐4
 rectangle
 I
 used
 to
 locate
 the
 12.
 
Now
 I’ll
 use
 the
 same
 rectangle,
 but
 in
 another
 position.”
 
 

Rotate
 the
 rectangle
 and
 again
 place
 it
 in
 the
 upper
 left-­‐hand
 corner.
 Again,
 lift
 the
 lower
 right-­‐
hand
 corner
 and
 write
 12
 in
 the
 square.
 Do
 the
 same
 for
 the
 2-­‐by-­‐6
 and
 the
 1-­‐by-­‐12
 rectangles,
 
writing
 12
 in
 the
 four
 additional
 squares.
 

Demonstrate
 the
 process
 again
 using
 the
 rectangles
 for
 the
 number
 9,
 showing
 that
 rotating
 the
 3-­‐
by3
 rectangle
 doesn’t
 matter
 since
 the
 lower
 right-­‐hand
 corner
 will
 be
 the
 same
 square
 either
 
way.
 

Invite
 the
 students
 to
 return
 to
 their
 seats
 and
 follow
 this
 process
 for
 each
 of
 their
 rectangles
 that
 
would
 fit
 on
 the
 squared
 paper.
 They
 can
 use
 the
 rest
 of
 class
 to
 finish.
 


 
DAY
 FOUR:
 INVESTIGATING
 PATTERNS
 ON
 OUR
 MULTIPLICATION
 TABLES
 

Engagement
 (10
 minutes):
 Ask
 students
 to
 take
 a
 look
 at
 their
 squared
 paper
 and
 the
 chart
 they
 
are
 creating.
 Does
 anyone
 recognize
 it?
 
 
 

 
Exploration
 (40
 minutes):
 Discuss
 the
 patterns
 in
 what
 they
 have
 done.
 Look
 at
 rows
 with
 patterns
 
they
 are
 familiar
 with
 …
 2s,
 5s
 and
 10s.
 Model
 how
 you
 continue
 to
 fill
 in
 the
 rest
 of
 each
 row
 and
 
column.
 Some
 students
 may
 also
 know
 the
 3s.
 You
 can
 show
 them
 how
 to
 continue
 skip
 counting
 
using
 a
 calculator,
 pressing
 3
 then
 +,
 then
 =
 repeatedly
 until
 that
 row
 and
 column
 are
 filled
 in.
 
Invite
 the
 class
 to
 go
 back
 to
 their
 desks
 and
 fill
 in
 the
 rest
 of
 the
 numbers
 themselves.
 Also
 tell
 
them
 that
 as
 they
 fill
 in
 their
 tables
 you
 want
 them
 to
 make
 note
 of
 any
 special
 patterns
 on
 special
 
3”
 x
 11”
 strips
 of
 paper.
 
 

 
Explanation
 (15
 minutes):
 When
 everyone
 has
 finished,
 post
 and
 compare
 what
 the
 students
 have
 
found.
 Some
 of
 the
 patterns
 will
 include:
 

In
 even
 numbered
 rows
 and
 columns,
 all
 of
 the
 products
 are
 even
 numbers.
 

In
 the
 odd
 numbered
 rows
 and
 columns,
 the
 products
 are
 odd,
 even,
 odd,
 even,
 odd,
 even.
 

In
 the
 5
 row
 and
 column,
 the
 products
 end
 in
 5,
 0,
 5,
 0,
 5,
 0.
 

For
 the
 10x
 column,
 you
 just
 have
 to
 add
 a
 0.
 

Everything
 in
 the
 11
 row
 and
 column
 has
 a
 double
 digit.
 

In
 the
 nines
 row
 and
 column,
 all
 of
 the
 products
 add
 up
 two
 nine.
 

Plus
 many
 more!
 


 

18

 
DAY
 FIVE:
 INVESTIGATING
 MORE
 PATTERNS
 ON
 OUR
 MULTIPLICATION
 TABLES
 

 
Engagement
 (10
 minutes):
 Pass
 out
 several
 sheets
 of
 multiplication
 tables
 to
 each
 student
 and
 ask
 
them
 to
 get
 out
 their
 colored
 pencils
 or
 crayons.
 Tell
 them
 that
 today
 they
 are
 going
 to
 investigate
 
even
 more
 patterns
 on
 the
 multiplication
 table.
 
 

Begin
 by
 modeling
 the
 multiples
 of
 6.
 
 

“First
 I
 need
 to
 make
 a
 list
 of
 the
 multiples
 of
 6.
 Read
 them
 to
 me
 from
 the
 6
 row
 or
 column
 of
 your
 
multiplication
 table.”
 (The
 list
 will
 go
 up
 to
 72.)
 Now
 demonstrate
 how
 you
 will
 cross
 off
 the
 
number
 6
 wherever
 it
 occurs
 on
 the
 chart,
 then
 the
 number
 12
 wherever
 it
 occurs,
 and
 so
 on.
 

“What
 is
 the
 largest
 number
 on
 the
 12-­‐by-­‐12
 table?”
 (144)
 “So
 we
 need
 to
 continue
 the
 list
 of
 
multiples
 to
 get
 as
 close
 to
 144
 as
 we
 can.
 Let’s
 add
 6
 to
 72
 to
 get
 the
 next
 number
 (and
 so
 on).”
 
 

“We
 could
 continue
 adding
 6s
 or
 we
 could
 use
 a
 calculator.
 Do
 you
 think
 we
 will
 land
 exactly
 on
 
144?
 Is
 144
 a
 multiple
 of
 6?”
 Invite
 students
 to
 explore
 their
 thinking
 out
 loud.
 


 
Exploration
 (40
 minutes):
 Now
 invite
 the
 students
 complete
 what
 you’ve
 started
 on
 the
 multiples
 
of
 6
 chart
 in
 their
 small
 groups,
 then
 the
 multiples
 of
 the
 ten
 remaining
 numbers
 (2-­‐5
 and
 7-­‐12)
 –
 
making
 sure
 to
 use
 separate
 charts
 for
 each
 number.
 
 

“As
 we
 did
 here,
 you’ll
 want
 to
 first
 list
 the
 multiples
 of
 the
 number,
 then
 color
 in
 all
 of
 the
 
multiples
 of
 that
 number
 on
 a
 fresh
 multiplication
 table.
 Be
 sure
 to
 color
 in
 every
 square
 for
 that
 
multiple.
 For
 example,
 for
 multiples
 of
 6,
 we
 crossed
 off
 all
 four
 6s
 that
 occurred
 on
 the
 chart
 and
 
all
 six
 12s.
 Continue
 until
 you
 have
 colored
 in
 all
 of
 the
 multiple
 squares
 and
 see
 what
 patterns
 
emerge.”
 

As
 the
 children
 work,
 write
 the
 numbers
 2-­‐12
 on
 the
 board
 leaving
 room
 underneath
 each
 so
 
group
 representatives
 can
 post
 sample
 charts
 for
 discussion
 when
 everyone
 is
 done.
 

 
Explanation
 (15
 minutes):
 Discuss
 the
 students’
 findings
 during
 the
 last
 15
 minutes
 of
 class.
 
Example
 questions
 for
 their
 consideration
 include:
 What
 did
 you
 notice?
 Which
 of
 the
 numbers
 
have
 just
 stripes?
 We
 colored
 in
 the
 multiples
 of
 only
 two
 square
 numbers,
 4
 and
 9.
 What
 did
 you
 
notice
 about
 them?
 

 
Evaluation:
 
 Monitor
 student’s
 oral
 and
 written
 responses
 to
 assess
 understanding
 of
 factor
 
patterns
 that
 emerge
 on
 the
 multiplication
 table.
 Also,
 ask
 the
 students
 to
 respond
 to
 this
 prompt
 
in
 their
 math
 journals:
 What
 do
 you
 know
 about
 7
 x
 6?
 

 

 
References:
 
 
Burns,
 M.
 (1987).
 A
 Collection
 of
 Math
 Lessons:
 From
 Grades
 3
 Through
 6.
 Sausalito,
 CA:
 Math
 Solutions.
 
Burns,
 M.
 (1991).
 Math
 By
 All
 Means:
 Multiplication
 Grade
 3.
 Sausalito,
 CA:
 The
 Math
 Solutions
 Publications.
 

 

19

Name __________________________________
EXPLORING OUR RECTANGLES
1. Which numbers have only one rectangle? List them from smallest to largest.

2. Which rectangles have a side with two squares on them? Write the numbers from
smallest to largest.

3. Which rectangles have a side with three squares on them? Write the numbers from
smallest to largest.

4. Do the same for rectangles with four squares on a side.

5. Do the same for rectangles with five squares on a side.

6. Which numbers have rectangles that are squares? List them from smallest to
largest. How many squares would there be in the net larger square you could
make?

7. What is the smallest number that has two different rectangles? Three different
rectangles? Four?


 

 

From A Collection of Math Lessons: From Grades 3 through 6. (c)1987 Math Solutions.

20

Lesson
 6:
 Billy
 Wins
 a
 Shopping
 Spree!
 

Grade
 Level
  Introduction:
 In
 this
 lesson,
 students
 will
 solve
 a
 real-­‐world
 problem
 –
 Billy
 Wins
 a
 
Shopping
 Spree
 –
 using
 their
 growing
 knowledge
 of
 multiplication,
 demonstrating
 
Third
 and
 Fourth
  that
 they
 understand
 both
 the
 meaning
 of
 and
 practical
 use
 for
 multiplication.
 

 
 

  Preparation:
 Divide
 the
 class
 up
 into
 groups
 of
 two
 to
 four
 students
 and
 distribute
 
Time
 Needed
  materials
 accordingly.
 
 

 
50
 minutes
  GLCE:
 3.N.MR.03.09
 Use
 multiplication
 and
 division
 fact
 families
 to
 understand
 the
 

  inverse
 relationship
 of
 these
 two
 operations,
 e.g.,
 because
 3
 x
 8
 =
 24,
 we
 know
 that
 

  24
 ÷
 8
 =
 3
 or
 24
 ÷
 3
 =
 8;
 express
 a
 multiplication
 statement
 as
 an
 equivalent
 division
 
Materials
  statement.
 
 

Copies
 of
 “Billy
  3.N.MR.03.10
 Recognize
 situations
 that
 can
 be
 solved
 using
 multiplication
 and
 
Wins
 a
 Shopping
  division
 including
 finding
 "How
 many
 groups?"
 and
 "How
 many
 in
 a
 group?"
 and
 
Spree”
  write
 mathematical
 statements
 to
 represent
 those
 situations.
 
 
worksheet
 
 
 
Engagement
 (10
 minutes):
 Tell
 the
 class
 that
 Billy
 is
 a
 fortunate
 boy
 who
 won
 a
 $25
 
Writing
 pencils
  shopping
 spree
 at
 the
 Science
 Museum
 Store.
 They
 will
 find
 a
 list
 of
 the
 items
 that
 

  he
 can
 purchase
 and
 the
 price
 for
 each
 item
 on
 their
 worksheet.
 
 
Explain
 that
 Billy
 can
 spend
 up
 to
 $25
 on
 any
 selection
 of
 the
 listed
 items.
 If
 he
 

  doesn’t
 spend
 the
 entire
 amount,
 he
 can’t
 keep
 the
 change,
 instead
 he
 will
 have
 a
 
store
 credit
 that
 he
 can
 use
 later.
 He
 can’t
 spend
 more
 than
 $25
 and
 cannot
 use
 
any
 other
 money
 that
 he
 might
 have
 …
 or
 ask
 his
 parents
 for
 some.
 They
 do
 not
 
need
 to
 calculate
 any
 sales
 tax.
 

Draw
 a
 model
 of
 the
 receipt
 on
 the
 board:
 


 

Science
 Museum
 Store
 
 
 
Receipt
 
 

___
 items
 @
 $3.00
  $__________
 

___
 items
 @
 $3.00
  $__________
 

___
 items
 @
 $3.00
  $__________
 

Total
  $__________
 

Store
 Credit
  $__________
 


 
Instead
 of
 duplicating
 blank
 receipts
 for
 the
 students
 to
 fill
 in,
 have
 them
 prepare
 
their
 own.
 This
 experience
 will
 help
 them
 learn
 how
 to
 organize
 their
 work
 on
 
paper.
 
 

They
 need
 to
 record
 Billy’s
 transaction
 two
 different
 ways:
 
 

1)
 
 In
 words,
 describing
 what
 he
 bought,
 how
 much
 each
 item
 cost,
 the
 total
 
amount
 he
 spent
 and
 the
 amount
 of
 any
 store
 credit
 he
 can
 use
 later;
 and
 
 

2)
 
 On
 the
 receipt
 that
 they
 prepare.
 

21

 
Exploration
 (25
 minutes):
 Invite
 the
 students
 to
 “shop”
 for
 Billy,
 writing
 their
 transactions
 both
 
ways.
 

 
Explanation
 (15
 minutes):
 Use
 class
 discussion
 to
 have
 some
 of
 the
 children
 present
 different
 ways
 
they
 found
 to
 spend
 exactly
 $25.
 This
 will
 reinforce
 the
 idea
 that
 problems
 can
 have
 more
 than
 
one
 solution.
 

 
Extension:
 Find
 the
 different
 combinations
 of
 $3,
 $4
 and
 $5
 items
 that
 equal
 exactly
 $25.
 When
 
students
 search
 for
 solutions
 by
 trial
 and
 error,
 they
 get
 great
 deal
 of
 number
 practice.
 Make
 sure,
 
however,
 that
 they
 understand
 the
 focus
 on
 the
 number
 of
 items
 at
 a
 particular
 price,
 not
 the
 
section
 of
 particular
 items.
 For
 example,
 buying
 five
 Koosh
 balls
 is
 the
 same
 solution
 as
 buying
 
three
 Koosh
 balls,
 an
 inflatable
 world
 globe,
 and
 a
 dinosaur
 model
 kit.
 In
 each
 case,
 Billy
 spends
 
$25
 buy
 buying
 five
 items
 @
 $5.
 

 
Evaluation:
 The
 students’
 written
 and
 oral
 responses
 will
 serve
 as
 a
 component
 of
 the
 summative
 
assessment
 of
 their
 understanding
 of
 multiplication,
 both
 its
 meaning
 and
 real-­‐world
 uses.
 
 
For
 their
 final
 Math
 Journal
 entry
 for
 the
 unit,
 invite
 them
 to
 respond
 to
 the
 prompt:
 


  “What
 I
 now
 know
 about
 multiplication.”
 

 

 
Reference:
 Burns,
 M.
 (1991).
 Math
 By
 All
 Means:
 Multiplication
 Grade
 3.
 Sausalito,
 CA:
 The
 Math
 Solutions
 
Publications.
 


 

22