VISUAL
REPRESENTATIONS
TO RAMP UP
INTERVENTIONS
By: Dina Mendola and Jamie Neary
“A strategy is most useful to students when it is theirs, built on and
connected to concepts and relationships they already own.”
(Van de Walle, 2004)
Background Information
Introductions
Cudahy School District Interventions
● Partnership with UW-Milwaukee and
the CORE Mathematics Project
● Add+Vantage Math Recovery for Tier
2 Interventions with Classroom
Teachers
● Math Recovery (MRIS) for Tier 3 with
Math Specialist
Learning Intentions and Success Criteria
We are learning how to…
●deepen our students’ fluency and understanding of single and double-digit addition
and subtraction.
We will be successful when we can …
●use visual models and color support to support the development of fluency with
single-digit addition and subtraction.
●distance the setting with screening and no color support to push students towards
their cutting edge of knowledge.
●clarify OA connections and expectations of fluency for addition and subtraction
basic facts, grades K - 2.
Building fluency K - 2
Working in tandem ~ OA and NBT
Counting and Cardinality/Operations
and Algebraic Thinking
Focus: Understand and use numbers
with meaning
Students gain experience and develop
proficiency in understanding how the
operations work and the relationship
between the operations. Properties,
meanings and uses of the operations
take center stage.
Number and Operations in Base Ten
Focus: Understand the structure of our
number system to compute
Students gain experience and develop
proficiency with computation strategies.
In grades K-2 these strategies are based
on their understanding of number, place
value, and properties of the operations.
Let’s start with the end in mind...
•Standard 2.OA.2:
Fluently add and subtract within 20 using mental
strategies. By end of Grade 2, know from memory
all sums of two one-digit numbers.
“Acquiring proficiency in single-digit arithmetic
involves much more than memorizing.”
(Adding It Up, NRC, 2001, p. 6)
What is “more”
and how do we
help our students
get there?
The awareness of the Standard Progressions is only
one aspect…
To be successful, the learning trajectory must consist
of instructional tasks or activities matched to each
level of thinking in a developmental progression!
Fluent to 5, 10, and 20 are the Standard Progressions but knowing when to
provide visual support (color) and when to distance the setting (screening and
flashing) will develop fact fluency with understanding.
With understanding, students can be successful with bare number tasks, as their
is a solid foundation of quantity!
Developmental Progressions
Addition and Subtraction within CCSSM (OA Domain)
Level 1: Count-three-times, Counting All or Taking Away
Level 2: Counting On / Counting down
Level 3: Flexibility with Numbers to Convert to an Easier
Problem
For students to become fluent in their facts, ultimately we
want them at a level 3. How do we get them there?
Four Types of
Number
Relationships
Used to Build
Fluency
Visual representations to
build these number relations
●
●
●
●
●
Finger Patterns
Dot Images
5-frames
10-frames
Reckenrack
How can color support, screening
and flashes advance thinking with
these visual representations?
Kindergarten
K.OA.3 Decompose numbers less than or equal to 10
into pairs in more than one way, e.g., by using objects
or drawings and record each decomposition by a
drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
K.OA.4 For any number from 1 to 9, find the number
that makes 10 when added to the given number, e.g.,
by using objects or drawings, and record the answer
with a drawing or equation.
Fluent to 5
but strategies
to solve
within 10.
Finger patterns ~ Anchor to 5 / Fluency to 10
● Fire Away
● Show two ways to make ____
o Examples: 5, 7, 8
● Doubles up to 10
● Show ___ add one more. How many?
● Bunny Ears
Dot Images
How many dots?
How did you see it?
How did you find
the total amount?
Dot Images
How many dots?
How did you see it?
How did you find
the total amount?
Dot Images
How many dots?
How did you see it?
How did you find
the total amount?
Dot Images
How many dots?
How did you see it?
How did you find
the total amount?
Dot Images
How many dots?
How did you see it?
How did you find
the total amount?
Dot Images
How many dots?
How did you see it?
How did you find
the total amount?
Dot Images
How many dots?
How did you see it?
How did you find
the total amount?
Five and Ten Frames show relationships of
small numbers
Five frames with color support
4 + 1 = 5
How many dots?
How did you see it?
How did you find
the total amount?
Five frames without color support
4 + 1 = 5
How many dots?
How did you see it?
How did you find
the total amount?
How many dots?
How did you see it?
How did you find
the total amount?
How many dots?
How did you see it?
How did you find
the total amount?
How many dots?
How did you see it?
How did you find
the total amount?
First Grade
Now that we have them fluent to 5 and exposed to 10s, we want them to:
Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.
Use strategies such as:
●
counting on
●
making ten
o
●
decomposing a number leading to a ten
o
●
Ex: 13 - 4 = (13 - 3) - 1 = 10 - 1 = 9
using the relationship between addition and subtraction
o
●
Ex: 8 + 6 = (8 + 2) + 4 = 10 + 4 = 14
Ex: knowing that 8 + 4 = 12, one knows 12 - 8 = 4
creating equivalent but easier or known sums
o
Ex: adding 6 + 7 by creating the known equivalent
o
6 + 6 + 1 = 12 + 1 = 13
Fluent to 10
but strategies
to solve within
20!
Moving beyond counting by ones
● Make a ten
● Use an easier problem
● Use doubles
● Anchor to 5
● Using helping fact
8+6
Put 8 counters on your first
frame & 6 counters on your
second frame.
Strategies:
•Make a ten
•Use a double
•Use fives
Make a ten:
8+6
How could you
make a 10?
Use a
Double:
8+6
What doubles
might you use?
Use fives:
8+6
Can you see
some fives?
DECOMPOSE TO 10:
15 - 6
Place 15 counters on
the double ten frame.
Fill the first frame up and
then place remaining on
the second frame.
How can you remove 6
counters in parts by
decomposing it in a way
that gets you to or “leads
to a ten”?
DECOMPOSE TO 10:
15 - 6
Remove 5 to get
to 10.
Then remove one
more.
What other
subtraction facts
would lend well to
this?
How to begin to transition to bare
number tasks
●Decompose numbers with frames and building equations
Represent with drop down notation
●Flashcards to apply strategies
8+7  Have students respond with strategies “Take 2 from the 7, so it is
10 and 5; 15.”
Bead Rack / Rekenrek
•It is comprised of two strings of 10 beads each,
strategically broken into groups of five.
•The structure of the rekenrek offers a visual for young
learners, encouraging them to “see” numbers within other
numbers… to see and use visual anchors of 5 and 10.
•For example…
How many to get to 10? So how many to get to 20?
Doubles +/- 1 or 2
Second Grade
Now that they are fluent to 10 and using strategies to solve within 20,
we want them to:
Add and subtract within 100, demonstrating fluency for
addition and subtraction within 20.
Use strategies such as:
●counting on
●making ten or getting to the “friendly” number
o
Ex: 42 + 9 = (42 + 8) + 1 = 50 + 1 = 51
●decomposing a number leading to a ten or to that
“friendly” number
o
Ex: 54 - 6 = (54 - 4) - 2 = 50 - 2 = 48
Fluent to 20 but
strategies
within 100.
Using Bead Strings
Provides a visual representation separating 5 and 10s.
Expands beyond Rekenrek’s representation of 20 beads.
Allows students to work on separating their place values and
understanding of what values make a number.
(45 = 4 tens and 5 ones or 40 + 5)
Linear representation that lends well to parallel
written tasks with an Empty Number Lines (ENL)
*Sentence strips work great!
Empty Number Lines
Without proper use and
questioning, any tool
(no matter how good) can become
a procedure, including ENL.
How can we change that?
ENL for addition
● Provide a ENL with a starting number. Have them jot
sum of each jump.
● Predictions → Visualize the jumps in their head without
writing to form their predicted answer, then solve to see if
they are accurately using mental math
o Helps students see errors they have in their thinking
and positively reinforces their mental math confidence.
● Change the unknown → Helps students see the
relationship between the numbers and their actions
How to move away from sticks and dots?
Resorts back to a Count
Three Times strategy
So how do we moveVisual representation does
NOT match the problem
students beyond this?
Students easily lose track of
their stick and dot drawings
Is the fundamental
foundation of incrementing /
decrementing by 10s and
1s present?
Distance the setting with addition and
subtraction!
Screening of the first quantity with stick support
By screening the first quantity of sticks and dots, we are pushing
students to think about mentally jumping those tens and ones
Teaching Split-Jump with notations with bare numbers
Example on the next slide
Notating the
Split-Jump
Strategy
Subtraction
Addition
Resources to share
TNCore - Common Core State Standards, Fluency in Mathematics
http://tncore.org/sites/www/Uploads/2.25.13Additions/fluency%20documents%20final.p
df
Core Math Partnership: UW-Milwaukee
http://uwm.edu/education/research/centers/cmser/core-math/
REKENREK
●“Using the Rekenrek as a Visual Model”
http://bridges1.mathlearningcenter.org/media/Rekenrek_0308.pdf
●mathrack.com - resources and video lessons on using Rekenrek
●Rekenrek activities: http://www.k-5mathteachingresources.com/Rekenrek.html
●Learning to Think Mathematically With a Rekenrek
https://cesa5mathscience.wikispaces.com/file/view/Learning+to+Think+Mathematically+with+
the+Rekenrek.pdf
Thank you!
Dina Mendola
[email protected]
Jamie Neary
[email protected]