Longwood University
Professional Development
Seminar
Algebra, Number Sense, and
Mathematical Connections
in Grades 3-5
Community of Learners
 Complete 3 X 5 notecard:
 Name
 Email
 School name and location
 Grade you teach
 Number of years teaching & grade levels
 Why are you interested in this workshop?
 Workshop Introductions
Course Materials
 Distribute Course Texts and Packets
 Your grant committment: 3-day summer workshop, 1-
day fall and 1-day spring workshops, a lesson plan due
by end of 2012-2013 school year, & a sampling of
classroom observations
 New 2009 VA SOL
 Patterns, Functions, and Algebra
 Number and Number Sense
 Computation and Estimation
 Complete Workshop Pre-Assessment
Resources
 Blanton, Maria, et. al. Developing Essential
Understanding of Algebraic Thinking: Grades 3-5.
Reston, Va.: National Council of Teachers of
Mathematics, 2011
 Jacob, Bill, and Catherine Twomey Fosnot. The
California Frog-Jumping Contest: Algebra. Portsmouth,
NH: Heinemann, 2007
 Russell, Susan Jo, Deborah Schifter, & Virginia
Bastable. Connecting Arithmetic to Algebra.
Portsmouth, NH: Heinemann, 2011
Resources
 Fosnot & Jacob. Young mathematicians at work:
Constructing Algebra. Portsmouth, NH:
Heinemann, 2010.
 Blanton, Maria. Algebra and the Elementary
Classroom: Transforming Thinking, Transforming
Practice. Portsmouth, NH: Heinemann, 2008
 Fosnot, C.T., and Dolk, M. Young mathematicians
at work: Constructing number sense, addition, and
subtraction. Portsmouth, NH: Heinemann, 2001.
Goals of the Workshop
1. To know more about algebraic thinking than you
expect your students to know and learn.
2. An awareness of different models and
representations to enhance algebraic thinking.
3. To become familiar with the connections
between number and operation concepts and
algebra concepts.
Goals of the Workshop
4. To know what mathematics to emphasize and
why in planning & implementing lessons.
5. To anticipate, recognize, and dispel students’
misconceptions about algebra.
6. Build on prior grades’ algebra ideas and know
later-grade connections  vertical alignment.
From NCTM Principles and Standards:
The Teaching Principle
“Effective mathematics teaching requires
understanding what students know and need to
learn and then challenging and supporting them to
learn it well… (p.16).
Teaching about algebraic thinking requires
knowledge that goes “beyond what most teachers
experience in standard preservice mathematics
courses” (p.17).
--- NCTM (2000). Principles and Standards for
School Mathematics. Reston, VA: Author.
Class Structure and Norms
 What do you need to do to
contribute to the learning of the group?
 What do you need to do to make
sure this is a learning experience for
yourself?
Think-Pair-Share
 What is algebra?
 Make a list of what you think might be ‘essential
components’ of algebra in grades 3-5 and grades
6-8. How are your lists the same and different?
 What classroom practices can teachers use to
support students’ development of algebraic
thinking?
Algebraic Reasoning is…
 The study of relationships among quantities.
 The study of structures.
 A set of rules and procedures.
 Generalized arithmetic.
--- Bassarear, T. (2005). Mathematics for Teachers.
Algebraic Thinking
“When children are given the chance to structure
number and operation in their own way, they see
themselves as mathematicians and their
understanding deepens.
They can make sense of algebra not as a funny set of
rules that mixes up letters and numbers handed
down by the authority of thousands but as a
language for describing the structure and
relationships they uncover” (p. 169).
---Fosnot & Jacob (2010).
Something to Consider
“What would you think about literacy instruction if
the goal was primarily for students to learn to spell?
Would you agree that simply teaching children to spell
provides them with the necessary skills to write in
various literary genres or read and critique the work of
others? In a similar way, mathematics instruction that
focuses only on arithmetic facts, skills, and procedures
– while these are important – leaves a big void in
developing children’s mathematical understanding”
(p. 83).
--Blanton, 2008.
Classroom Practices…
5 Process Standards:
 Problem Solving
 Reasoning and Proof
 Communication
 Connections
 Representation
---Principles and Standards for School
Mathematics (2000) and Virginia Standards of
Learning (2009)
Virginia Department of Education
Instructional Goals that foster Algebraic Thinking
 “Represent: Provide multiple ways for children to
systematically represent algebraic situations.
 Question: Ask questions that encourage children to
think algebraically.
 Listen: Listen to and build on children’s thinking.
 Generalize: Help children develop and justify their
own generalizations” (p. 94).
--Blanton, 2008.
Generalizing…
 “Children should view building a generalization as an
important mathematical activity. It is at the heart of
algebraic thinking” (p. 116).
--Blanton, 2008.
 “Generalizing is the process by which we identify
structure and relationships in mathematical
situations.”
 “Generalizations may be expressed in a number of
ways—through natural language, through algebraic
notations using letters as variables… or, even through
tables and graphs.”
---Blanton, et al., 2011, p.9.
Task: Chair and Leg Problem
 Using at least two different strategies, show all your work
and solve the following problem:
Suppose that you have some chairs, and each chair has 4
legs. How would you describe the relationship between
the number of chairs and the corresponding number of
chair legs?
--Blanton, et al., 2011, p.9
 Discuss with a partner how you
solved the problem.
Generalizing the Chair and Leg Problem
 Language: “The number of chair legs is four times the
number of chairs.”
 Algebraic notation: l = 4c, where c represents the
number of chairs and l represents the number of legs.
 Function table: see p.10, Blanton et al.
 Graph: see p.10, Blanton et al.
---Blanton, et al., 2011, pp. 9-10
Using Word Problems:
The Role of Context
 The potential to model the situation is built in the
problem (Fruedenthal, 1973).
 The situation allows learners to realize what they are
doing.
 The situation prompts learners to ask questions,
notice patterns, wonder, ask why and what if. (p. 43)
---Fosnot, C.T., and Dolk, M. (2001).
Hunter’s model train is set up on a
circular. Six telephone poles are
spaced evenly around the track. The
engine of Hunter’s train takes 10
seconds to go from the first pole to the
third pole. How long will it take the
engine to go all the way around the
track once?
NCTM and VA Process Standard:
Making Connections
“Understanding
algebraic thinking
deeply requires you not only to know
important mathematical ideas but also
to recognize how these ideas relate to
one another.”
---Blanton,
et al., 2011, p. 3.
Our Focus: 5 Big Ideas
1.
The properties of arithmetic used with the four operations are the
same for algebra.
2.
An equation uses an ‘equal sign’ to show that two quantities are
equivalent.
3.
A variable is a tool that describes relationships in many different
ways.
4.
Quantitative reasoning extends relationships between and among
quantities.
5.
Functional relationships are expressed in words, symbols, tables,
or graphs, and reasoning occurs between these representations.
---Blanton,
et al., 2011, pp. 12-14.
Teaching Responsibilities
“We have a critical, very important role to play as teachers.
We walk the edge between the structures of mathematics
and the development of the child. This means we have to
understand thoroughly the development of the mathematics
by considering the progression of strategies, the big ideas
involved, and the emergent models that potentially can
become powerful forms of representations with which to
think.”
--- Fosnot and Jacob, 2010, p. 25.
Big Idea 1
Addition, subtraction, multiplication,
and division operate under the
same properties in algebra as they do
in arithmetic.
---Blanton, et al., 2011, p. 15.
Essential Understanding 1a
 The fundamental properties of number and
operations govern how operations behave and
relate to one another.
--Blanton et al., p. 20
What properties are used in real
number computations?
 If you add two real numbers in either order you have the
same result.
 If you multiply two numbers and then multiply by a third,
or if you multiply the first by the product of the second and
third, you have the same result.
What properties are used in real
number computations?
 If you multiply a real number by the sum of two real
numbers, or if you add the products of the first real
number with each of the second and third, you have the
same result.
 What properties do zero and one have with respect to all
other real numbers?
Fundamental Properties of
Number and Operations
Task 1
 For each operation, make a list naming the
properties students need to learn and know for
grades 3-5.
 Provide an example of each using real numbers.
 Use variables to represent the properties.
Where are the different properties stated in
the Virginia Standards of Learning?
 Task 2: Compare your lists of properties to those stated in the
VA SOL found in your packet. In which grade do students focus
on each property?
 Task 3: Compare your lists of properties to those stated on page
16 in the Blanton et al. course text.
 Task 4: For each of the properties on page 16 in the Blanton et
al. course text develop a series of arithmetic tasks that would
help children generalize that property. How would children
express this generalization in words and in symbols?
 Why are these properties missing for subtraction and division?
Justify your thinking!
Frog Jump Training Camp
http://www.youtube.com/watch?v=3ctSME2uEIg
Make your own origami frog!
With a partner, use an open number line to
demonstrate 3 jumps and 2 jumps is the same as 2
jumps and 3 jumps.
Can you think of another example using the
commutative property? Share.
Demonstrate the associative
property of addition with an open
number line using your origami
frog.
Essential Understandings
 The fundamental properties are essential to
computation. (Essential understanding 1b)
--Blanton et al., p. 17
 Simplifying algebraic expressions entails
decomposing quantities in insightful ways.
(Essential understanding 1d)
 As teachers, our goal is to use ‘relational thinking’ and
understand:
 How properties are used in computations
 How to explicitly identify them
--Blanton et al., p. 20
 Think ‘mental math’ strategies
Task: Using Mental Mathematics
1. First, pencils or pens down, solve the given
problems mentally in at least 2 different ways.
2. Second, write down and record your thinking –
the strategies you used.
3. Share and discuss your strategies with a partner.
4. Now that we have shared our strategies, are any of
the strategies generalizable? If so, write the
generalizations in words and/or symbols.
Children’s Invented Strategies
 Analyze each child’s strategy used to solve:
28 + 35
 What is the logic behind each strategy?
 Are each of these useful strategies? Why or why
not?
Analyzing Student Work
“The general ideas you notice in your students’
work become the starting points for engagement in
generalizing about the operations.”
Teacher’s role:
 1st step – observe and notice
 2nd step – help students share what they notice
 3rd step – planning opportunities for generalizing
---Russell, Schifter, & Bastable, (2011), pp. 7, 9, & 16.
Three Standards for Assessing Students’
Algorithms
 Efficient enough to be used regularly without
considerable loss of time and frustration due to the
number of recorded steps required
 Mathematically valid
http://www.youtube.com/watch?v=PgjGXHZXCuY
 Generalizable
Generalizable Strategies
for Addition and Subtraction
 Compensation: making user-friendly numbers; add an
amount to one addend and subtract the same amount
from the other addend; making an easier equivalent
expression; 29 + 6  add 1 and subtract 1; 29 + 6 = 30 + 5
 Symbolic Generalization: a + b = (a + 1) + (b – 1)
 ‘Shift in the same direction’ – add the SAME amount to
both numbers: 128-19 ‘is the same as’ 129-20; use an
‘open number line’ model.
 Symbolic Generalization: a – b = (a + 1) – (b + 1)
A Focus on Reasoning
“Gradually, students learn to develop and
justify mathematical arguments based on
images and models. Asking questions that
require reasoning about problem without
computing requires students to think more
deeply about the operations.”
---Russell, Schifter, & Bastable (2011), p. 19.
2.3
Other Addition Thinking Strategies
1. Splitting – partial sums; decomposing
numbers using place value (i.e., expanded
form); also called “1010” method in the Dutch/
Netherlands curriculum
2. Making Jumps of Ten – called the “N10”
method, split or decompose only one number
(use hundred chart or open number line)
---Fosnot, C.T., and Dolk, M. (2001).
Partial Sums for Addition Algorithm
25
+ 34
9
50
59
20 + 30 = 50
5+ 4= 9
59
Model with base 10
blocks…
Draw an open
numberline…
Partial Sums for Addition Algorithm
This works for larger numbers too!
305
+ 627
900
20
12
932
300 + 00 + 5
600 + 20 + 7
900 + 20 + 12
= 932
Model with base 10
blocks…
Draw an open
numberline…
Other Subtraction Thinking Strategies
1. Splitting – partial differences; decomposing numbers using
place value (i.e., expanded form); also called “N10” method
2. Making Jumps of Ten – split or decompose only one
number; called the “N10” method in the Dutch/ Netherlands
curriculum.
3.
Count-back from the minuend. (use hundreds chart)
4. Count-up from the subtrahend – ‘adding up.’
5. Use four-fact families: relate the inverse operations of
addition and subtraction.
---Fosnot, C.T., and Dolk, M. (2001).
Splitting or Partial Differences for
Subtraction Algorithm
548
- 233
300
10
5
315
Model with base 10
blocks…
Draw an open
numberline…
Splitting or Partial Differences for
Subtraction Algorithm
642
- 293
Model with base
10 blocks…
Draw an open
numberline…
500 + 130 + 12
- 200 - 90 - 3
300
+ 40
________ +9
= 349
‘Adding Up’ Subtraction Algorithm
(Context: think making change with money)
50
- 23
Model with base
10 blocks…
Use hundreds
chart…
Draw an open
numberline…
What do we need to add to
23 to get 50?
Count-on from 23 until we
reach 50.
+2
+ 10
+ 10
+ 5
27
Activity: A Multiplication Strategy
What Do You Know About ________
“Asking what students know about an expression
can be a good way to start the year because it invites
a range of student observations and provides a
window into the understandings and experiences
students bring to the classroom.
It is also useful throughout the year because you
can choose expressions that connect to whatever
ideas students are currently studying.”
---Russell, Schifter, & Bastable (2011), p. 16.
More Generalizations
“Generalizations in arithmetic can be derived from
the fundamental properties” (p. 21).
--Blanton et al., 2011.
“Generalizing arithmetic includes helping children
see, describe, and justify patterns and regularities
in operations on and properties of real numbers”
(p. 12).
---Blanton, Maria (2008).
4 Approaches to Justifying the
‘Truth of a Mathematical Statement’
1.
Accepting the claim on authority.
2. Trying it out with examples.
3. Applying mathematical reasoning based on a visual
representation or story context.
4. Proving using algebraic notation and the laws of
arithmetic.
---Russell, Schifter, & Bastable (2011), pp. 52-53.
Arithmetic leads to Generalization
1. Choose accessible numbers when asking students to
investigate general ideas about the operations.
2. Ask students to think about an expression or a set of
related expressions without carrying out the
computation.
3. Ask students to show their ideas using cubes,
number lines, arrays, story contexts, or other
representations.
4. Keep a list of ideas in progress.
---Russell, Schifter, & Bastable (2011), pp. 19 – 21.
Arithmetic leads to Generalization
5. The teacher can then ask students to pose a conjecture, a
statement about the pattern or regularity the students are
noticing that can be either true or false.
6. These conjectures can be investigated to determine if
they are always true, or sometimes true and need to be
revised.
7. These conjectures can then be written as generalizations
with words and/or symbols once they are found to be
true.
Forming a Generalization
 Are the following statements true?
3+0=3
0+4=4
100 + 0 = 100
 How do you know these are true?
 Can you write a conjecture in words?
 Is there a way you can show that your conjecture is true
or false? How?
 If your conjecture was true, write it as a generalization
in words and in symbols.
Another Generalization
 Are the following statements true?
14 x 1 = 14
1 x 359 = 359
1 x 2,346 = 2,346
 How do you know these are true?
 Can you write a conjecture in words?
 Is there a way you can show that your conjecture is true
or false? How?
 If your conjecture was true, write it as a generalization
in words and in symbols.
Activity:
Commutative Property of Addition
“Consider the commutative property of
addition for yourself. How many ways
can you explain why 2 + 5 = 5 + 2? Use
number lines, story contexts, drawings,
and a variety of representations.”
---Russell, Schifter, & Bastable (2011), p. 51.
Show Me… Convince Me
The Associative Property of Addition and The
Associative Property of Multiplication
“What if Ms. Raymond or Ms. Perez comes in
the room and says she doesn’t understand, or
she doesn’t even trust what you are saying.
How could you convince her that this
equality is true? How can you prove it is
true? You can use drawings, cubes, or a story
to convince her.”
---Russell, Schifter, & Bastable (2011), p. 18.
Generalizing patterns with Even/Odd
 Develop a set of addition problems that helps children
think about what happens when they add two even
numbers. Is the result even or odd? How do you
know? How could you show your result symbolically
using algebra?
 Develop a set of subtraction problems that helps
children think about what happens when they subtract
one odd number from another odd number. Is the
result even or odd? How do you know? How could
you show your result symbolically using algebra?
Helping Children to Generalize
Arithmetic
Sample Teacher Questions
12 + 18 =
What do you notice about the addends?
(They are even).
22 + 18 =
What do you notice about your
solutions? (They are even)
28 + 14 =
What can you conjecture about the sum
of two even numbers? (An even number
plus an even number is always even).
34 + 26
Do you think this is always true? Why?
---Blanton, Maria (2008). (p. 16).
What if a conjecture is false?
 A counterexample is a case where a conjecture does
not hold true.
 For example, if a student conjectured that the sum of
an even number and an odd number is even; the single
counterexample, 4 + 5 = 9, would be sufficient to prove
the conjecture is false.
 It is usually not possible to test all possible cases so
there is always some sense of uncertainty about a
conjecture.
Building on Generalizations
Use the generalization to reason instead of needing to calculate
Will the following sum be even or odd? Why?
1895 + 198 7 + 2073 + 5999
When students reason from structure instead of calculation
they are reasoning algebraically.
---Blanton, Maria (2008). (p. 17-18).
Investigating Relationships with the Hundred Chart
Students can analyze the structure of a series of moves
 Commutative of addition: (down 1 row and right one
column is the same as right one column and down 1
row) 15 + 10 + 1 = 15 + 1 + 10
 Addition and Subtraction are Inverse operations:
Adding a number and then subtracting that number
results in a change of zero(down 1 row and then up 1
row) 15 + 10 – 10 = 15
Why do we need parentheses?
 Shane had a $50 bill. He bought a t-shirt for $10 and a
pair of jeans for $25. How much change should Shane
get back when he pays? (We are assuming the prices
already include sales tax.)
 First we need to find out how much he spent….

10 + 25 = 35
 Then we need to find out how much he gets in change

50 – 35 = 15
 He will receive $15 in change.
How could we combine these two
operations in one math sentence?
 If we were to write our processes in words in one
sentence we could say…..
 Money paid – Total price = Change received
 I could write
 50 – (10 +25) = 15
 Parentheses help us write one math sentence that has
more than one operation.
 When we have parentheses the order of operations says
that we must do the calculation inside the parentheses
first.
More practice with using parentheses
 Suzanne bought a pair of gloves for $12 and a scarf for
$32. How many glove and scarf pairs can Suzanne buy
for $220?
 Describe your processes in words…what do we need to
do to solve this problem?
 Find the cost of one scarf-glove pair

12 + 32 = 44
 How many $44 sets can be bought with $220?

220 ÷44 = 5
 I can buy 5 scarf-glove pairs
 How can you use one math sentence to describe these
processes?

220 ÷ (12+32)
Parentheses and the Distributive Property
 Hannah loves lollipops. She has 3 bunches of cherry
lollipops with 5 lollipops in each bunch. She has 3
bunches of orange lollipops with 6 lollipops in each
bunch. How many lollipops does Hannah have?
 Describe in words your process




Cherry lollipops + Orange lollipops = Total lollipops
Find the cherry lollipops
3 x 5 = 15
Find the orange lollipops
3 x 6 = 18
Find the total lollipops
15 + 18 = 33
 Written using one math expression


3 x 5 + 3 x 6 = 33
3(5 + 6) = 33
Why is 3 x 5 + 3 x 6 = 3(5 + 6)
 Set model
Cherry
Can you see 3 sets of 5 and 3 sets of 6?
Can you see 3 sets of 5 + 6?
Orange
Why is 3 x 5 + 3 x 6 = 3(5 + 6)
 Array or area model
Cherry
Orange
Can you see 3 rows of 5 and 3 rows of 6?
Can you see 3 rows of 5 + 6?
Thinking Strategies…
Develop Algebraic Thinking
“If students early work with arithmetic consists
solely of practicing standard algorithms and
memorizing facts to use in those algorithms, they
are unlikely to engage in algebraic thinking during
this study.
However, when students’ learning of arithmetic
focuses on reasoning about mathematical
relationships, they naturally engage in a great deal
of algebraic thinking while thinking about
arithmetic” (p. 23)
---Blanton, et al., 2011
Homework
 Read:
 Blanton et al. (2011): Foreword, Preface,
Introduction, and beginning part of Chapter
1 (pp. 1-24)
 Jacob & Fosnot (2007): Day One, (pp. 15-20)
 Do:
 Develop a set of subtraction problems that
helps children think about what happens
when they add two odd numbers. Is the
result even or odd? How do you know? How
could you show your result symbolically
using algebra?