Multiplication and Division
Small Group Instructional Approaches and Games
Multiplication and Division Levels
Level 1: Initial
Grouping
Level 2:
Perceptual
Counting in
Multiplies
Level 3:
Figurative
Composite
Grouping
Student “gets” groups but has to count by 1s and relies on the visible objects.
Student does not attend to the structure of the groups when counting.
Shows some multiplicative counting, but needs the visible objects to count.
Could attend to the group structure, but needs visible group markers (the card, the
container, the dot marking the row of the array). Uses composite thinking to help
determine the total.
Student is able to re-present the pieces.
They are unable to coordinate the count for the number of groups when they are
trying to determine how many two’s are in twelve.
Level 4:
Repeated
Abstract
Composite
Grouping
Level 5:
Multiplication
and Division as
Operations
Students at Level 3 understanding the iteration (a composite of) items but cannot
coordinate the appropriate number of iterations. 3, 6, 9 – the 3 becomes
coordinated with the 6 and the 9.
Students is able to use the composite unit a number of times (repeated addition or
subtraction) without visible markers for the groups and individual items.
“Dual dial” keeping track of the counts and the markers. Keeping track of the
groups, the pieces, and the total in the absence of materials.
Student uses figural re-presentation (using a spatial pattern) for the groups on the
marble task and a motor re-presentation (motor/tapping) on the banana task.
Quickly derive facts and recall known facts.
Student is able to coordinate two composites up there.
Student is able to talk about number relationships in terms of the context of the
problem and in a formal math sense.
Demonstrates that relational knowledge of multiplication and division.
Repeated addition and subtraction is no longer used.
©Created by Jenni Scholla
Resources: Purple, Green, and Red Multiplication Books as well as other resources.
Construct 0-1
Goal: multiplication as a process; equal groupings
Helps students who…
FNWS and BWNS Need fluency with skipby multiples
counts
Things In Groups
0
Grouping Items
0
Materials Needed:
Notes:
Numeral Track (see later These students probably are not
for more ideas)
attaching a lot of quantity to the
verbal sequence, but it’s something
SMART Board or Small
you can continually work on. When
Group
working with small groups, add the
dot strips to help add the quantity.
Make lists of things that come in 2s,
3s, 4s, and so on. Draw their
attention to “groups” of things.
Create lists.
Any manipulative
Provide many opportunities of
students to group items. “here are
16 counters. How can we group
them equally?
Other questions: How many legs at
your table? How many eyes for
these 4 people? Etc.
©Created by Jenni Scholla
Resources: Purple, Green, and Red Multiplication Books as well as other resources.
Construct 1-2
Goal: lead to multiplicative counting for visible and screened items.
Activity
A
Lemonade Stand
Helps students who...
Direct modelers
Constructs 0-2
Materials Needed:
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Rolling Groups
Direct Modelers
p. 16-19 in AVMR Constructs 0-2
Book 2 Teacher’s
Handbook
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Loops and
Groups/Circles
and Stars
Dot Cards (strips
and dice
patterns)
Lemonade Stand
Game board
cubes and cups
dice
Watch for strategy use: CB1 or
other CB
Rolling Group
Game boards and
Materials
Rolling Groups
record sheet
dice
Watch for strategy use: CB1 or
other CB
Direct Modelers
Constructs 0-2
Direct Modelers who
need the visual support
yet
Construct 2
Notes:
Watch for strategy use: CB1 or
other CB
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dice
Recording sheet
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dot strips
SMART Board dot
strips and numeral
track
(helps see the connection between
repeated addition and
multiplication)
Can be done whole group with
SMART Board doc or small. Ask:
How many cards? How many dots
on each? How many all together?
Vary the order of questioning. This
can lead into symbolic
representations with number
sentences.
Dot Arrays
(unscreened for
lower constructs,
partial screens
and flashed, for
higher
constructs)
Construct 2-4(depends on
the multiple being used
and how much is
screened)
Students who visual
support

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arrays are located
on the shared
folder
Origo (see Jenni if
interested - these
are organized by
strategy)
Make sure to tie in both
multiplication and division with
these to
GREAT for strategy work
(i.e.doubling, building up/building
down)
©Created by Jenni Scholla
Resources: Purple, Green, and Red Multiplication Books as well as other resources.
Helps provide a visual
structure and
organization before
moving onto number and
verbal sequences
Circle and Stars/r Unscreened to Screened
Loops and
Groups
(screened and
flash
MR Purple p.
163)


SMART doc or you
could make on
paper with
card/lids
Be creative – birds
in trees, cookies on
a plate, etc.
Things to watch for: How are they
counting these? (starting from one,
stress-counting, skip-counting)
Ask: How many groups? How many
in each? How many all together?
Construct 3
Goal: Getting them from direct modeling to using additive subtractive strategies (repeated addition or
subtraction). Working towards full screened – no visible markers (bare tasks).
Construct 4
Goal: Can solve all screened task using additive subtractive strategies (repeated addition or subtraction).
These students would benefit from relational tasks now.
Using Multiplication Language Ideas in Class
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Are ten 8s the same as eight 10s. How do you know?
Fluently double a number – doubling up to 100
Fluently half a number – halving numbers up to 100
How are 10 facts related to 5 facts?
Using arrays to show 8 x 3 = 6 x 4
How can I use 9 x 8 to help me solve 18 x 4? (Construct 4)
What non-count by strategies an I use to solve 8 x 6? (Use array model and students can find and partition out
5s, etc.
How can I use 2 x 8 = 16 to solve 4 x 8?
If 14 x 3 = 42, what is 15 x 3 =?
Commutative Property – “flip flops”
Inverse Property
Multiplication Property of One
Multiplication Property of Zero
©Created by Jenni Scholla
Resources: Purple, Green, and Red Multiplication Books as well as other resources.
Thoughts to Ponder: Strategy vs. Numeral Sequences
Problem solving strategies are flexible and work efficiently in a variety of settings and extend beyond the basic
facts. For example: If I know that x5 is half of x10, when presented with a problem 243 x 5, I can do 243 x 10 which
is 2430 and then I can half that (half of 2000 is 1000, half of 400 is 200, half of 30 is 15) so mentally I can calculate
243 x 5 as 1, 215.
Numeral Sequences are a skill, and important and useful skill worth teaching. But, when taking it out of the
context of basic facts, skip counting to solve 243 x 5 is not an efficient strategy. It is important to keep in mind that
there is a distinction between merely saying the sequence of multiples and the conceptual understanding of using
them to count items in repeated groups.
When planning instructional activities and games, be cognizant of the purpose of the activity and how it will extend
beyond basic facts. Make sure a variety of instructional approaches are used and support verbal skip-counting
sequence by providing visual/concrete experiences with them to help students understand the quantity.
Remember: fact fluency will not really be truly be established until students have had other strategies (direct
modeling and counting)
Goal: Understanding multiplication as repeated addition and using multiplicative counting
Class/Group Count
Arounds
Helps students who...
All levels really – Helps
them hear the verbal
sequence of the multiples
Numeral Track
Treasure Hunt
Numeral sequence
fluency
Trio for Multiples
Purple MR Book
Numeral sequence
fluency (multiple before
and after)
Numeral sequence
fluency (multiple before
and after)
Quick Draw
AVMR Teacher Handbook
Course 2 p. 14
Materials Needed
Numeral tracks for desired
multiple; SMART Board
files or paper versions)
Treasure Hunt Cards
3-4 sets of desired
multiples
3-4 sets of desired
multiples
©Created by Jenni Scholla
Resources: Purple, Green, and Red Multiplication Books as well as other resources.
Notes:
Considered mixing up
sets (i.e. doing one row of
2s and one row of 4s for
doubling and halving; 3s
and 6s; 2s, 4s, and 8s)
Goal: Understanding multiplication and division as inverse operations and building fluency (as soon as students
are moving to equal groups, start incorporating that relationship).
Helps students
who…
Materials
Needed
Notes
*You can adapt many
multiplication games to
also be division games
by merely altering how
you play. Be creative.
Multiplication/Division
Inverse Concentration
Online Games: Division
http://www.fun4thebrain.com/division.html
Multiplication Spiral
(good
multiplication/division
practice in general)
Playing cards
Salute
Numeral or
playing cards
http://www.fun4thebrain.com/division.html
Dice
Place Markers
Array Cards
©Created by Jenni Scholla
Resources: Purple, Green, and Red Multiplication Books as well as other resources.