Properties of Operations
& Whole Number
Computation
Deborah Schifter (EDC) & Susan Jo Russell (TERC)
Teachers Development Group
March 2015
Projects from which our examples are
drawn come from more than a decade of
collaboration of:
Deborah Schifter (EDC), Susan Jo Russell (TERC),
Virginia Bastable (Mathematics Leadership Programs
at Mount Holyoke College),
and many collaborating teachers and their students.
Funded in part by the National Science Foundation
Today’s Session
•  Part One: Why explore the properties of the operations?
•  Part Two: What does this work look like in elementary
classrooms?
•  Part Three: Concluding comments
Part One:
Why explore the properties of the
operations?
Solve using two different strategies
47 + 28
15 × 18
47 + 28
= 40 + 7 + 20 + 8
= 40 + 20 + 7 + 8
= 60 + 15
= 75
Commutative property of addition
7 + 20 = 20 + 7
a+b=b+a
47 + 28
= 47 + 20 + 8
= 67 + 8
= 75
Associative Property
47 + 28
= 47 + (20 + 8) = (47 + 20) + 8
a + (b + c) = (a + b) + c
47 + 28
47 + 28 = 45 + 30 = 75
47 + 28 = 50 + 25 = 75
Additive Identity
47 + 28 =
(47 + 28) + (3 – 3) =
(47 + 3) + (28 – 3) =
50 + 25 = 75
Common Errors: 47 - 28
47
-28
21
What’s important about studying the
behavior of the operations?
•  Understanding the procedures requires
understanding the structures that underlie the
procedures.
•  Common procedural errors stem from not
understanding those structures.
•  Examining structure brings coherence to the
study of computation.
Common Core State Standards
•  Grade 1: Understand and apply properties of operations
and the relationship between addition and subtraction.
•  Grade 2: Use place value understanding and properties of
operations to add and subtract.
•  Grade 3: Understand properties of multiplication and the
relationship between multiplication and division.
•  Grade 4: Use place value understanding and properties of
operations to perform multi-digit arithmetic.
•  Grade 5: Apply and extend previous understandings of
multiplication and division to multiply and divide fractions.
Footnote
Elementary students need not hear or use formal terms
for these properties.
Part Two:
What does this work look like in
elementary classrooms?
Part Two:
What does this work look like in
elementary classrooms?
Grade 2: Commutative property of addition. What about
subtraction?
Grade 3: Equivalent addition expressions and the additive
identity. What about subtraction?
Grades 1 and 3: Associative property of addition. What
about multiplication?
Grade 3: Distributive property of multiplication over addition
47
- 28
21
“You don’t put any more on and you don’t take any away,
so the sum stays the same.”
What happens if you change the order in
a subtraction expression?
17 – 9 = 8
9 – 17 ???
What happens if you change the order in
a subtraction expression?
9 – 17 = 0
What happens if you change the order in
a subtraction expression?
9 – 17 = -8
9 - 17
“invisible numbers”
60 – 50 = 10
50 – 60 = invisible 10
47
-28
“40 – 20 is 20, 7 – 8 is . . . No, that won’t work.
47 – 20 is 27, then subtract 8, that’s 19.”
Commutative Property of Addition:
Video Example: Grade 2
20 + 5
1 + 24
19 + 6
6 + 19
23 + 2
2 + 23
175 + 266
266 + 175
What happens with subtraction?:
Video example grade 2
7–3=4
3–7=
•  The teacher asks students to explicitly consider and
articulate how these addition expressions are behaving.
•  Once they have articulated their “rule,” she asks them to
explain, to make an argument.
•  They use a representation as the means to explain why
the generalization must be true.
•  Although students examine examples with specific
numbers, they begin to talk about how the operation
behaves in general: “if you just keep switching them
around . . . you’re not taking away or adding”
The importance of comparing operations
•  Through examining subtraction, the students are
challenged to explain and construct arguments.
•  Examining what happens with subtraction leads the
students to consider the limits of their generalization—that
commutativity is a property of addition but not subtraction.
Why is a focus on the behavior of the
operations important?
It is a key place in the curriculum where students can:
•  Notice regularities (MP8)
•  Develop conjectures about those regularities (MP8)
•  Articulate their ideas with precision (MP6)
•  Represent structure (MP7)
•  Construct arguments (MP3)
What do you notice?
2 + 8 = 10
3 + 7 = 10
4 + 6 = 10
5 + 5 = 10
6 + 4 = 10
7 + 3 = 10
27 + 15 = 42
30 + 12 = 42
33 + 9 = 42
36 + 6 = 42
39 + 3 = 42
42 + 0 = 42
Equivalent Addition Expressions:
Grade 3
addition
When we have an ^ expression, we change the numbers
sum
but still have the same answer.
The numbers can go up or down.
We change the numbers by making one less and the other
one bigger.
from one of the addends
to the other addend
We can take away one ^ and then add one ^.
We could switch a 3.
30 + 53
33 + 50
9 + 4 = 10 + 3 What is happening when you change the expression 9 + 4 to 10 + 3? Does this work with other numbers? Does this work with all numbers? How do you know?
Does the same thing happen with
subtraction?
10 + 3 = 13
9 + 4 = 13
8 + 5 = 13
7 + 6 = 13
10 – 3 = 7
9–4=5
8–5=3
7–6=1
Representation-based Argument
•  In the elementary grades, the tools available to students
to develop arguments are representations, including
physical models, pictures, diagrams, and story contexts.
•  The use of such representations makes the structure of
the operations visible.
•  The representations provide referents for the components
of the generalization and the relationships among them.
•  The representation becomes a means for communication
among students.
Equivalent subtraction expressions
17 – 5 = 12
18 – 6 = 12
19 – 7 = 12
.
.
.
.
Representing Equivalent Subtraction
Expressions
17 – 5 = 12
18 – 6 = 12
19 – 7 = 12
....
1) Picture of model made with interconnecting
cubes.
2) Story context: A boat can only hold 12 people. If
more people get on, the extras have to get off. If 17
people get on, 5 have to get off. If 18 people get
on, 6 have to get off, etc.
3) Story context: Cookies
Representing equivalent subtraction
expressions—
Video example Grade 3
17 – 5 = 12
18 – 6 = 12
19 – 7 = 12
.
.
.
.
What’s important about studying the
behavior of the operations?
•  Content: through representing multiple examples and
contrasting operations, students see the underlying
structure of properties and behaviors of the operation
•  Practices: students notice and articulate regularities
across examples and use representations to construct
arguments about what they are noticing
•  Engagement: Students are expected to and are given
time to articulate their own ideas and engage with other
students’ ideas. Students’ reasoning is central—there is
something significant to reason about
What happens when you change an
addend?: Video example grade 1
What happens when you change an
addend?
•  9 + 10 = 19
•  So, 9 + 9 must be equal to 18.
•  9 + 9 = ?
•  9 + (9 + 1) = (9 + 9) + 1
Associative property of addition
Opening Lesson
7 + 5 = 12
7+6=
7 + 5 = 12
8+5=
9 + 4 = 13
9+5=
9 + 4 = 13
10 + 4 =
What do you notice?
What’s happening here?
Opening Lesson
7 + 5 = 12
7 + 6 = 13
7 + 5 = 12
8 + 5 = 13
9 + 4 = 13
9 + 5 = 14
9 + 4 = 13
10 + 4 = 14
What do you notice?
What’s happening here?
Grade 3 conjecture
If you add 1 to an addend and keep the
other addend the same, the sum increases
by 1.
Adding 1 to an addend: Video example
grade 3
•
Phases of working on a generalization
•  Work with small numbers.
•  Articulate a general claim.
•  Demonstrate the claim with a context or representation.
•  Use the representation to make a general argument.
•  Consider what happens with a different operation.
15 × 18
•  Distributive property
a. (10 + 5) × 18
(10 × 18) + (5 × 18)
180 + 90
270
b. 15 × (10 + 8)
(15 × 10) + (15 × 8)
150 + 120
270
•  Multiplicative identity
(15 × 2) × (18 ÷ 2)
30 × 9
270
15 × 18
DISTRIBUTIVE PROPERTY
15 × (10 + 8) = (15 × 10) + (15 x 8)
= 150 + 120
(10 + 5) × 18 = (10 × 18) + (5 x 18)
= 180 + 90
= 270
= 270
COMMON ERRORS
(10 + 5) × (10 + 8) = (10 × 10) + (5 × 8) = 140
15 × (10 + 8) = 150 + 8 = 158
What about multiplication?: Adding 1 to a factor
Writing prompt. In a multiplication problem, if
you add 1 to a factor, I think this will happen
to the product:
What about multiplication?: Adding 1 to a factor
•  Come up with a representation or story context for 7 × 5 =
35.
•  What changes in your representation or story of 7 × 5 to
show what happens when 7 increases by 1? (7 + 1) × 5
•  What changes in your representation or story of 7 × 5 to
show what happens when 5 increases by 1? 7 × (5 + 1)
Adding 1 to a factor:
Video example grade 3
•
Contrasting addition and multiplication
7 + 5 = 12
7 + 6 = 13
7 + (5 + 1) = (7 + 5) + 1
Associative property of addition
7 × 5 = 35
7 × 6 = 42
7 × (5 + 1) = (7 × 5) + (7 × 1)
Distributive property of multiplication over addition
Contrasting addition and multiplication:
Video example grade 3
•
Part Three:
Concluding Comments
Why do properties of operations matter in
the elementary classroom?
•  Content—underlie all computation strategies; lack of deep
understanding of properties/behaviors leads to errors
•  Practices—noticing regularity (MP8), articulating with
precision (MP6), conjecturing (MP8), constructing
arguments (MP3), and making use of structure (MP7)
•  Connection between arithmetic and algebra—deep
understanding of how operations behave is critical when
students are operating with algebraic symbols
•  Engagement—familiar numbers and operations make it
possible for students to enter these ideas; students
respond to being asked to reason about significant
mathematical content
Resources
•  https://www.illustrativemathematics.org/MP3
•  http://www.mathedleadership.org/ccss/itp/operations.html
•  Connecting Arithmetic to Algebra, published by
Heinemann
•  Developing Mathematical Ideas professional development
series, new edition to be published by NCTM beginning
fall 2015; in the meantime, get ordering info from
http://www.mathleadership.org