OPERATIONS AND ALGEBRAIC THINKING - OA K-2
PUTTING IT TOGETHER:
NUMBER STRATEGIES AND
STORY PROBLEM TYPES
Math Teacher Leader Meeting
January 10, 2012
Beth Schefelker
Connie Laughlin
Lee Ann Pruske
Hank Kepner
Mathematics – Grade K
Table 1 Addition and Subtraction Situations.
Add to
Take from
Put Together / Take
Apart
Compare
Result Unknown
Two bunnies sat on the grass. Three
more bunnies hopped there. How
many bunnies are on the grass now?
2+3=?
Change Unknown
Two bunnies were sitting on the
grass. Some more bunnies hopped
there. Then there were five
bunnies. How
many bunnies hopped over to the
first two?
2+?=5
Start Unknown
Some bunnies were sitting on the grass.
Three more bunnies hopped there.
Then there were five bunnies. How
many bunnies were on the grass
before?
?+3=5
Five apples were on the table. I ate
two apples. How many apples are on
the table now?
5–2=?
Five apples were on the table. I ate
some apples. Then there were three
apples. How many apples did I eat?
5–?=3
Total Unknown
Three red apples and two green apples
are on the table. How many apples are
on the table?
3+2=?
Addend Unknown
Five apples are on the table. Three
are red and the rest are green. How
many apples are green?
3 + ? = 5, 5 – 3 = ?
Some apples were on the table. I ate
two apples. Then there were three
apples. How many apples were on the
table before?
?–2=3
Both Addends Unknown
Grandma has five flowers. How many
can she put in her red vase and how
many in her blue vase?
5 = 0 + 5, 5 = 5 + 0
5 = 1 + 4, 5 = 4 + 1
5 = 2 + 3, 5 = 3 + 2
Difference Unknown
(“How many more?” version):
Lucy has two apples. Julie has five
apples. How many more apples does
Julie have than Lucy?
Bigger Unknown
(Version with “more”):
Julie has three more apples than
Lucy. Lucy has two apples. How
many apples does Julie have?
Smaller Unknown
(Version with “more”):
Julie has three more apples than Lucy.
Julie has five apples. How many apples
does Lucy have?
(“How many fewer?” version):
Lucy has two apples. Julie has five
apples. How many fewer apples does
Lucy have than Julie?
2 + ? = 5, 5 – 2 = ?
(Version with “fewer”):
Lucy has 3 fewer apples than Julie.
Lucy has two apples. How many
apples does Julie have?
2 + 3 = ?, 3 + 2 = ?
(Version with “fewer”):
Lucy has 3 fewer apples than Julie. Julie
has five apples. How many apples does
Lucy have?
5 – 3 = ?, ? + 3 = 5
DECEMBER HOMEWORK
How will you implement this information into your work in
the next month? Be prepared to share your successes and/or
challenges at the January MTL meeting.
Work with a teacher or teacher team
 Explore textbooks for story problem
formats
 Try problem types with a group of
students

LEARNING INTENTIONS
Understand
strategies that can be
used to solve various problem
situations.
Understand
strategies that can be
used to develop number fluency.
SUCCESS CRITERIA
 You
will be successful when you can
identify and implement number
strategies that students can use to
develop number fluency and solve
story problem situations.
WHAT IS FLUENCY?
 What
does fluency mean to you?
 Read
the last paragraph on pg. 18 and 19.
 What
is the “gist” of fluency as outlined in
the Progressions for the Common Core
State Standards in Mathematics?
BUILDING FLUENCY AND INSTRUCTIONAL
IMPLICATIONS
 “So
the important press toward fluency
should also allow students to fall back on
earlier strategies when needed. By the
end of the K-2 grade span, students have
sufficient experience with addition and
subtraction to know single-digit sums
from memory…
p. 19 0A Progressions document
BUILDING FLUENCY AND INSTRUCTIONAL
IMPLICATIONS
 “So
the important press toward fluency
should also allow students to fall back on
earlier strategies when needed. By the
end of the K-2 grade span, students have
sufficient experience with addition and
subtraction to know single-digit sums
from memory… this is not a matter of
instilling facts divorced from their
meanings, but rather as an outcome of a
multi-year process that heavily involves
the interplay of practice and reasoning.”
p. 19 0A Progressions document
MOVING FROM CONCRETE TO ABSTRACT
REPRESENTATIONS
 Direct
modeling, Counting on and
Numeric Reasoning
 Use
the chart on pg. 36 and your
problem type chart to make sense of
the summary reading on pg 20-21.
 How
are the problem types related to
student strategies?
HOW DOES OA GROW?
OPERATIONS AND ALGEBRAIC THINKING K-2
K5
Grade 1
Grade 2
Responsible for 4
problem types
Responsible for 8
problem types
Responsible for 12
problem types
Work within 5,
then 10
Work within 20
Work within 100
Physically act out
problems
Count on
Make 10
Decompose through 10
Doubles/Near doubles
Concrete models
to show all parts
Semi-concrete models
Matching with objects
Tape diagrams
Abstract model
Drawings show
quantities
HOW DOES OA GROW?
OPERATIONS AND ALGEBRAIC THINKING K-2
K5
Grade 1
Grade 2
Responsible for 4
problem types
Responsible for 8
problem types
Responsible for 12
problem types
Work within 5,
then 10
Work within 20
Work within 100
Physically act out
problems
Count on
Make 10
Decompose through 10
Doubles/Near doubles
Concrete models
to show all parts
Semi-concrete models
Matching with objects
Tape diagrams
Abstract model
Drawings show
quantities
LEARNING INTENTIONS
Understand
strategies that can be
used to solve various problem
situations.
Understand
strategies that can be
used to develop number fluency.
GRADE 1: CONTENT STANDARD 1.OA.6
Add and subtract within 20, demonstrating fluency
for addition and subtraction within 10.
Use strategies such as
• counting on
• making ten
• decomposing a number leading to a ten
• using the relationship between addition &
subtraction
• creating equivalent but easier or known sums
STRATEGIES THAT BUILD
SINGLE DIGIT ADDITION FLUENCY
Counting
Make
Use
on.
a ten.
an easier “equivalent” problem.
Use doubles
Use fives
Use a 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.
Use some other
equivalent problem.
COUNT ON: 8 + 6
Write an equation.
8+1+1+1+1+1+1=14.
MAKE A TEN: 8 + 6
How could you
make a ten?
Move 2 counters
to the top frame.
Then you have 10
and 4 more counters.
Write an equation.
8 + 6 = 8 + 2 + 4 = 10 + 4 = 14
USE A DOUBLE: 8 + 6
What doubles
might you use?
USE A DOUBLE: 8 + 6
What doubles
might you use?
Reason 6 + 6 = 12;
then add 2 more.
Write an equation.
8+6=6+6+2=
12 + 2 = 14
USE FIVES: 8 + 6
Can you see some
fives? Where?
USE FIVES: 8 + 6
Can you see some
fives? Where?
Reason: 5 + 5 is 10;
need to add 3 more
and 1 more.
Write an equation.
8+6=5+5+3+1
= 10 + 4 = 14
STANDARDS FOR MATH PRACTICE
Which Standard for Math Practice were
you using as you worked through these
activities?
1. Make sense of problems and persevere in
solving them.
3. Construct viable arguments and critique
the reasoning of others.
6. Attend to precision.
7+9
9+8
6+7

Select a problem.
 Tell an addition story.
 Draw a strategy card for the group.
 Everyone uses ten frames and counters to reason
through the strategy and writes an equation(s) that
shows the reasoning.
 Share, compare, and discuss as a group.
 Repeat with another strategy card for the same
problem or a new problem.
Reflect: Which strategies seem to
work best for each problem?
STANDARDS FOR MATH PRACTICE
Which Standard for Math Practice were
you using as you worked through these
activities?
1. Make sense of problems and persevere in
solving them.
3. Construct viable arguments and critique
the reasoning of others.
6. Attend to precision.
WHY A 10 FRAME?
How does modeling each strategy
with the same manipulative model
help develop student understanding?
Develops
number fluency and
flexibility
Anchors to 5 and 10
Place value fluency
AS YOU THINK OF YOUR WORK…
How
does this information
impact your work with teachers
of all grades?
…
with Special Education , ELL,
and older students who don’t
have these skill sets.
PROFESSIONAL PRACTICE
EXPLORING STRATEGIES
Number Strategies using Tens Frames:
 Work with a teacher or teacher team using
tens frames strategies
 Model problem types with a group of
students using tens frames
 Explore textbooks for number strategies
using tens frames
 Video tape your work with teachers and or
students working with ten frames and story
problem types.
LEARNING INTENTIONS
Understand
strategies that can be
used to solve various problem
situations.
Understand
strategies that can be
used to develop number fluency.
SUCCESS CRITERIA
 You
will be successful when you can
identify and implement number
strategies that students can use to
develop number fluency and solve
story problem situations.
FEEDBACK QUESTION:
(1) Story problem types; (2) number strategies; (3)
developmental learning progressions (from concrete to
abstract); and (4) instructional models (like ten
frames) are interrelated parts that must be cohesively
connected for students to develop both fluency and
number sense in the K-2 grades and the OA K-2
CCSS.
 Considering the 4 areas above, which areas do
you need more practice to develop your skill in
applying them with teachers and students?