```Kentucky Education Cooperatives
Conceptual Building Blocks Series
Day 2
“All Kids, All Successful, All the Time”
Day 2
• Addition and Subtraction Strategy Progressions
• Story Problems
• Place Value
• Identifying Error Patterns
• Center Activities
• Subtract C from F
Strategies
• Count all (two collections)
• Count on
• Count back/count down to/count up from
Developing Essential Understanding of Addition and Subtraction Pre-K – Grade 2, NCTM publication, page 78, Early Numeracy
Research Project (Clarke, 2001)
Strategies
• Basic strategies
• Derived strategies
• Extending and applying addition and subtraction
using basic, derived and intuitive strategies
NCTM publication, page 78, Early Numeracy Research Project (Clarke, 2001)
Combinations and Partitions
Activities
• Build fluency to 5
• Build fluency to 10
• Build Part-Part-Whole Understanding
• Combinations and Partitions of 20
Using:
Five frames, ten frames, linking cubes, domino
patterns, 20 frames
K.OA, K.NBT, 1.OA, 2.OA.3
Begin with Five!
Move to ten!
Think of a way to make 7
Begin building on and off from a
number
Part-Part-Whole and Whole-part-part
• Use of five, ten, and twenty frames with no
empty boxes. Show all partitions of the
whole.
• Use of five, ten, and twenty frames with
some empty boxes. Show all combinations of
the whole.
Use of Frames
Use of Frames
• Objectives
– Help students visualize doubles (e.g., 4+4; 6+6)
– Help students use doubles in computation
• The visualization is key:
1+1=2
2+2=4
3+3=6
4+4=8
Check Up
• Bunny Ears
One hand – to combine and partition within five
Two hands – to combine and partition within ten
Five and Ten Frames
Empty
Use counters
Check Up
• Ask the students to give the combinations and
partitions of numerals to five
What goes with three to make five?
If I have two, how many more do I need to
make five?
What two numbers combine to make five?
If I have two pencils, how many more do I
need to have four pencils?
For our struggling students
• Some students have difficulty in thinking
Use cubes that will fit on their finger tips
Use Velcro and structure
Use foam ten frames with removable dots
Use Wiki Sticks to assist in building the
symbols
Salute!
From Kentucky Center for Mathematics, Kentucky
Numeracy Project Intervention Guide
www.kymath.org
Subtraction Situations
The standards stress the importance of students
being able to use addition and subtraction in all
situations.
The Four main problem types are:
Take from
Put Together and Take Apart
Compare
(With unknown in all positions)
Problem Type
Result Unknown
Change Unknown
Start Unknown
Take From
Result Unknown
Change Unknown
Start Unknown
Put together and
take apart
Total Unknown
unknown
Compare
Difference
Unknown
Bigger Unknown
Smaller Unknown
We normally see problems that ask students to
“join” or “separate” to find the unknown part.
We tend to ask students to “put together” for
addition and “take from” for subtraction. These
definitions are limited and if these are the only
exposure students have, they will have difficulty
when the situation calls for something other
than “put together” or “take away”.
Take for example, the following problem:
Bob has 3 nickels and Bill has 7 nickels.
How many more nickels does Bill have than
Bob?
Problem Solving Mat
Story Problems
• Use of quantity in context assists students in
attaching meaning to quantity, as well as, the
actions of the operations
• Incorporate number talks along with the
story problems
• Develop problem strings along with the story
to assist students to develop efficient
strategies
K.OA.2, 1.OA.1, 2, 2.OA.1
Story Mats
Frog Story Mat
Gathering for Winter
Whole Group Activity
Place Value
• Conceptual place value develops as students
are involved in mental math strategies.
• Positional place value should not be taught in
a procedural memorized manner. It gets in
the way of a students development of a
conceptual understanding.
Developing Number Knowledge, 2012. Wright, Robert J., Ellemor-Collins, David., and
Tabor, Pam. Sage Publications.
Unitizing and Place Value
• Unitizing in place value is the understanding that ten
ones is the same one ten.
• Represents a HUGE shift in understanding
• 56 is 5 tens and 6 ones
and is also: 4 tens and 16 ones
3 tens and 26 ones
2 tens and 36 ones
1 ten and 46 ones
56 ones
Developing Number Knowledge, 2012. Wright, Robert J., Ellemor-Collins,
David., and Tabor, Pam. Sage Publications.
Place Value
• Grouping items
Cups and counters
Sticks and bundles
Dot strips
Ten frames
K.NBT, 1.NBT.2,3, 4, 5,6, 2.NBT.1, 2, 3, 4
A Conceptual Understanding of Ten
=
Developing Number Knowledge, 2012. Wright, Robert J., Ellemor-Collins, David., and Tabor,
Pam. Sage Publications.
Adding and Subtracting Tens to a
=
3 tens
or
30
Developing Number Knowledge, 2012. Wright, Robert J., Ellemor-Collins, David., and
Tabor, Pam. Sage Publications.
Adding and Subtracting Tens off the
and
23
and
“23… 33, 43, “43”
20
Developing Number Knowledge, 2012. Wright, Robert J., Ellemor-Collins, David., and Tabor,
Pam. Sage Publications.
Adding and Subtracting Tens and Ones
+
23
and
22
20, “30, 40, 43, 44, 45”
Developing Number Knowledge, 2012. Wright, Robert J., Ellemor-Collins, David., and Tabor,
Pam. Sage Publications.
Using Bundles and Sticks
300 (3 groups of hundreds), 20 (2 groups of tens), and 4 (4 sticks)
324
Developing Number Knowledge, 2012. Wright, Robert J., Ellemor-Collins, David., and Tabor, Pam.
Sage Publications.
7
+
5
What moves do you want to make?
Using Dots
300 (3 hundreds), 20 (2 tens) and 4 (4 ones) is 324
“Gretchen”
Video
Mental Math
• Solve the following:
43 + 28
65 - 27
1.NBT.4, .5, .6; 2. NBT.7 .8. 9; 3.NBT.2; 4. NBT.4
Student Work
Betty
43
+ 28
61
43
+ 28
711
Alice
“40 + 20 is 60
and 8 + 3 is 11
and 60 + 10 is
70, and one
more is 71”
Student Work
Lisa’s Work
65
- 27
42
Jason’s Work
65 – 27 = “37, 47, 57,
is 30, then 3 more is 60,
5 more is 33+5 is 38”
Common Misconceptions when
• “Subtract the smaller from the larger” is a rule that
children apply regardless of minuend or subtrahend.
62 – 45 = 23
• Not regrouping 34 + 28 = 52
• Ignoring the zeros
2 13
303
– 154
259
Instructional materials that support
children in using mental math strategies
• Use of composite units
Bundles and Sticks
Dot Strips
Frames
Dots on popsicle sticks
Unifix Cubes – Towers
Cups and counters
Base Ten Blocks
Thou Shall Not CARRY
38
+ 49
Strategies based on Place Value
38 + 49
38 is 30 + 8
49 is 40 + 9
30 and 40 make 70
8 + 9 make 17 which is 10 and 7, so 70 and 10 is
80 plus 7 is 87
• Use the split strategy to solve the following:
56 + 32
134 + 643
A Jump Strategy
38 + 49
49 + 10 is 59, plus 10 is 69, plus 10 is 79, then
one more is 80, then 7 more is 87
+10 +10 +10
+1
+7
______________________________________
49 59
69
79
80
87
• Solve the following using a jump strategy
64 + 33
132 + 54
Partial Sums
38
+ 49
70
17
87
136
+ 553
600
80
9
689
267
+ 841
1000
100
8
1108
Partial Sums Expanded Notation
533 + 327
500 + 30 + 3
300 + 20 + 7
800 + 50 + 10 = 860
• Solve the following using a partial sums
strategy
327 + 488
Why do so many students struggle
with subtraction?
 We teach them to “take away” or
borrow.
 Subtraction is neither commutative nor
associative
 Our sequence of learning is wrong
Place Value Strategy for Subtraction
56 – 27
56 is SPLIT apart into 50 and 6
27 is SPLIT apart into 20 and 7
56 – 20 = 36 and 36 – 7 = 29
• Solve the following using a split strategy
435 - 227
Jump Strategy for Subtraction
45 – 27
-2
-5 -10 -10
__________________________________
18
20
25
35
45
Sources:
• NCTM. 2011. Achieving Fluency: Special Education and Mathematics. Page 134,
136
• Walle, Van de. 2006. Teaching Student Centered Mathematics K-3. Page 185
• NCTM. 2011. Developing an Essential Understanding of Addition and Subtraction.
• Solve the following using a jump strategy:
56 - 38
Partial Differences
56
- 23
30
3
33
371
- 285
100
-10
-4
86
813
- 139
700
-20
-6
674
NCTM. 2011. Developing an Essential Understanding of Addition and Subtraction.
Pages 43-44
Compensation
34 + 29 (add one to 29 to make it thirty; add,
then subtract the one back off)
34 + 30 – 1 (64 - 1= 63)
• Subtraction by compensating
53 – 28 (add two to 28 to make thirty,
subtract, then take two back off)
53 – 30 + 2 = 23 + 2 is 25
Wright, Robert J., et. al. Developing Number Knowledge. 2012
• Solve the following problems using
compensation:
45 + 39
75 - 38
Transforming
58 + 27 (Add 2 to 58 and take 2 off 27;
maintaining the quantity of the entire problem)
60 + 25 = 85
• Subtracting by Transforming:
56 – 29 (add one to both numerals – keeping
the distance between the numerals the same)
57 – 30 = 27
• Solve the following using a transforming
strategy:
68 + 25
77 - 39
NCTM. 2011. Developing an Essential Understanding of Addition and Subtraction. Page 47.
• Use the adding ten strategy to solve the
following:
53 - 27
“Same Change” method used in Subtraction to Avoid
Regrouping
6000
- 3642
6000 - 1 = 5999
3642 - 1 = 3641
2358
46
- 28
46 + 2 = 48
28 + 2 = - 30
18
NCTM. 2011. Developing an Essential Understanding of Addition and Subtraction. Page 47
• Solve the following using the “same change”
strategy:
5000 – 2657
Identifying Error Patterns
A. 7 + 8 = 14
C. 7 + 6 = 12
B. 8 + 6 = 13
D. 8 + 5 = 12
Taken from Error Patterns in Computation, Using Error Patterns to Help Each
Student Learn by Robert B. Ashlock. 2010
Identifying Error Patterns
A. 56
+ 6
17
B. 18
+ 30
48
C. 8
+ 16
15
D. 42
+ 56
98
E. 85
+ 6
19
Taken from Error Patterns in Computation, Using Error Patterns to Help Each
Student Learn by Robert B. Ashlock. 2010
Identifying More Error Patterns
A. 32
- 16
16
B. 245
- 137
112
C. 524
- 298
374
D. 135
- 67
132
Taken from Error Patterns in Computation, Using Error Patterns to Help Each
Student Learn by Robert B. Ashlock. 2010
Reflection Activity
Center Activities for Day 2
•
•
•
•
•
•
•
What Number Is…?
Game of Chance
100 or Bust
Clear the Board
Make Ten Rummy
Real Counting On
Race to Twenty
Make and Take Day 2
• Frame Cards
Day 2 Reflection and Post-Assessment
• Three things I learned today are….
• Two things I will implement in my classroom
are…..
Complete the post-assessment
```