Longwood University Professional Development Seminar
Algebra, Number Sense, and
Mathematical Connections
in Grades 3-5
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
 Wickett, Maryann, et. al. Lessons for Algebraic
Thinking: Grades 3-5. Sausalito, CA: Math Solutions
Publications, 2002.
 Friel, Susan, et. al. Navigating through Algebra in
Grades 6-8. Reston, VA: National Council of Teachers
of Mathematics, ,2001.
Constructing Algebra
“Manipulation of numbers to produce an answer
can seem like a magic trick to learners if they
haven’t constructed the implicit relations for
themselves. The importance of this construction
cannot be overemphasized, because it is
precisely what enables learners to generalize” (p.
15).
----Fosnot & Jacob (2010). Young mathematicians at work: Constructing
Algebra. Portsmouth, NH: Heinemann.
The Magic Trick
 Choose any number (56)
 Add 3 (59)
 Multiply by 2 (118)
 Add 4 (122)
 Divide by 2 (61)
 Subtract the original number (61-56 = 5)
 The result is always 5
----Reimer (1992). Historical Connections in Mathematics: Resources
for Using History of Mathematics in the Classroom. Aims Education
Foundation.
Where is the magic?
 Choose a number
 Add 3
 Multiply by 2
 Add 4
 Divide by 2
 Subtract the number
 The result is
(x)
(x + 3)
2(x+3)
2(x+3) + 4
(x+3) + 2
x+3+2–x
5
Another trick
 Choose any number.
 Add 5.
 Double the result.
 Subtract 4.
 Divide by 2.
 Subtract the original number.
 Use algebra to show why the result is 3.
----Reimer (1992). Historical Connections in Mathematics: Resources
for Using History of Mathematics in the Classroom. Aims Education
Foundation.
Relational versus
Instrumental Understanding
Big Idea 4
Quantitative reasoning extends
relationships between and among
quantities to describe and generalize
relationships among these quantities.
---Blanton, et al., 2011, p. 39.
Conceptual Foundation for Algebra
“The experience of performing direct
comparisons with physical materials and
then representing them with algebraic
[symbolic] statements builds a conceptual
foundation that allows us to act on these
symbolic statements.
When symbolic or algebraic statements are
introduced with no specific context, little
meaning can be attached to them” (p. 43).
--Blanton et al., 2011
Essential Understanding 4b
“Known relationships between two
quantities can be used as a basis for
describing relationships with other
quantities” (p. 42).
--Blanton et al., 2011
Algebraic Substitution
 Substitution can be used to find a new relationship
between two quantities.
 10 apples weigh the same as 2 melons.
 How many melons would weigh the same as 25
apples? How do you know?
More Substitution
Substitution can be used to evaluate an
expression.
If M = 5, K = 8, and L = 14
Find the value of L + M – K
More Substitutions
 +  = 82
 - 10 = 58
What is the value of ?
How do you know?
X + Y = 53
Y – 14 = 36
What is the value of x?
How do you know?
Transitive Property
The transitive property says that if a =b and b = c,
then a = c.
2+=
 = 3 + 5 so…
2 +  = 3 + 5 so
What is the value of ? How do you know?
Open Sentences and
the Transitive Property
14 +z= y
y=7x7
What is the value of z? How do you know?
“The flexibility in representing equality relationships sets
the stage for solving equations” (Blanton, et al., p. 45)
 Using a Double Open Numberline
 Day Two: Jumping Buddies; Complete Appendix D
(Jacob & Fosnot, 2007, pp. 21-26)
 ‘Jump and step’ on classroom number line
 Draw ‘jumps and steps’ on a double open number
line for Appendix D activity
Using double open number lines
to represent & solve equations
 Day Six & Day Seven: The Frog-Jumping Contest;
Complete Appendix I (Jacob & Fosnot, 2007, pp. 4553)
 ‘Jump and step’ on classroom number line
 Draw ‘jumps and steps’ on a double open number
line for 3 different frogs: Sunny, Cal, and Legs (see
Appendix I activity)
Frog Jump Olympics
Divide into teams, with each team selecting
one of the following activities and record
your work on poster paper:
 Appendix J: “Pairs Competition”
 Appendix K: “Hitting the Mark”
 Appendix L: “The Toads’ Three-jump, Two-step
Event”
(Jacob & Fosnot, 2007, pp. 21-26)
Using Algebra Tiles
Now that you have solved these problems
with the open number line try it with the
Algebra tiles.

 A representation-based proof connects the
story to the symbolism. Using more than one
representation strengthens student
understanding of these connections.
Equivalence:
Relationships Between Quantities
“These ideas [using a variety of representations
or models] about the relationships of quantities
are important steps in developing the ability to
make sense of problems that have no story or
real-world context… without this relational
understanding of the equal sign, [students miss
looking for and reasoning about]… the value that
makes both quantities the same” (p. 45).
--Blanton et al., 2011
Big Idea 5
Functional thinking includes generalizing
relationships between covarying quantities,
expressing those relationships in words,
symbols, tables, or graphs, and reasoning
with these various representations to
analyze function behavior.
---Blanton, et al., 2011, p. 47.
From NCTM Principles and Standards:
The Teaching Principle
“Worthwhile tasks should be intriguing, with a
level of challenge that invites speculation and hard
work.
Such tasks often can be approached in more than
one way, such as using an arithmetic counting
approach, drawing a geometric diagram and
enumerating possibilities… tasks accessible to
students with varied prior knowledge and
experience” (p. 19).
--- NCTM (2000). Principles and Standards for School
Mathematics. Reston, VA: Author.
Algebraic Thinking
“Algebraic
thinking encompasses the set of
understandings that are needed to interpret the
world by translating information or events into the
language of mathematics in order to explain and
predict phenomena…”
--- Lawrence & Hennessy (2002). Lessons for algebraic
thinking: Grades 6-8. Sausalito, CA: Math Solutions
Publications.
Algebraic Thinking
Applying these understandings effectively often
requires the following components:
 Using or setting up a mathematical model, if
needed
 Gathering and recording data, if needed
 Organizing data and looking for patterns
 Describing and extending those patterns
 Generalizing findings, often into a rule
 Using findings, including any rules, to make
predictions” (p. xi).
- Lawrence & Hennessy (2002). Lessons for algebraic thinking:
Grades 6-8. Sausalito, CA: Math Solutions Publications.
--
Essential Understanding 5a
“A function is a special mathematical
relationship between two sets, where each
element from one set, called the domain, is
related uniquely to an element of the second
set” (the range) (p. 48).
---Blanton, et al., 2011, p. 48.
Are these functions?
1. {(father, children)}
2. {(Parker, his age)}
3. {(Jill, her height)}
Create your own set that is a function. Then
create your own set that is not a function.
--Collins, A. and Dacey, L. (2011). The Xs and Whys of
Algebra: Key Ideas and Common Misconceptions. Portland,
Maine: Stenhouse.
Piles of Tiles: A growing pattern that is a
function too!
Pile 1
Pile 2
Pile 3
6 blocks
10 blocks
14 blocks
Wickett, et. al., 2002, p. 197
Essential Understanding 5d
“In working with functions, several
important types of patterns or relationships
might be observed among quantities that
vary in relation to each other: recursive
patterns, co-variational relationships, and
correspondence rules” (p. 51).
---Blanton, et al., 2011
Two of Everything
 Written Lily Hong
Recursive Patterns
2, 6, 10, 14, 18, ….
 What type of pattern is this?
Growing pattern
 What is the pattern?
+4 pattern
 This is a recursive pattern because we use the previous
value to determine the next value in the pattern.
Recursive Patterns
with a Magic Pot
In
Out
1
3
2
5
3
7
4
9
5
+2
+2
+2
10
?
+2
Wickett, et. al., 2002, p. 12
Recognizing Co-variation
with the same Magic Pot
+1
+1
+1
+1
In
Out
1
3
2
5
3
7
4
9
5
10
?
+2
+2
+2
+2
Wickett, et. al., 2002, p. 12
How would you describe this pattern using
recursive thinking? Using co-variation?
It is still
difficult to
predict the
number of
coins that
will come
out when
100 coins
are put in.
In
Out
2
4
4
8
6
12
8
10
16
20
Wickett, et. al., 2002, p. 8
Shifting from Repeating Patterns
Toward Constructing Generalizations
Tabular (T-chart) approach: Numeric
a. Describe patterns in recursive relationships:
how outputs are related to one another (look
vertically, how change occurs from one stage to
the next stage); also called an iterative
relationship
b. Describe patterns in functional relationships:
how each input is related to the corresponding
output (look horizontally, find a rule based on
the stage or term); also called an explicit
relationship
---Driscoll, M. (1999). Fostering algebraic thinking: A guide for
teachers grades 6-10. Portsmouth, NH: Heinemann.
Correspondence and
Two of Everything
In
Out
1
2
2
4
3
6
4
5
8
10
Wickett, et. al., 2002, p. 8
Can you describe this pattern as a correspondence
between the In and Out values?
How
many
coins
come out
of this
magic pot
if 100
coins are
put in?
In
Out
1
3
2
5
3
7
4
5
9
11
How would you describe this pattern
using correspondence?
This kind of
In
Out
thinking
makes it
much easier
to predict
how many
coins will
come out of
the pot when
100 are put in
the pot.
2
4
4
8
6
12
8
10
16
20
Make your own magic pot rule
 Make your own table of values for a magic pot.
 Describe your pattern as a correspondence (also
called a function).
 Switch your pattern with a partner and see if they can
describe your pattern as a function.
Wickett, et. al., 2002, pp. 21-22
Essential Understanding 5e
“Functions can be represented in a
variety of forms, including words,
symbols, tables, and graphs” (p. 55).
---Blanton, et al., 2011
Task: Do the Following for all
three Banquet Table Patterns
 Examine each table pattern and extend the pattern to 4
table arrangements.
 Describe the pattern between the # of tables and # of
people in words.
 Make a table with the # of tables and # of people.
 Write an algebraic equation for the correspondence
between the number of tables and the number of people
 Use the correspondence to find the number of people
who can be seated around 50 triangular tables, 40 square
tables, and 35 hexagon tables.
Two Banquet table patterns:
Triangles and Squares
Wickett, et. al., 2002, p. 224-238
One More Banquet table pattern: Hexagons
Wickett, et. al., 2002, p. 239
Tables for Table Patterns
Triangle Table
Pattern
# of
# of
Tables
People
1
3
Square Table
Pattern
# of
# of
Tables
People
1
4
Hexagon Table
Pattern
# of
# of
Tables
People
1
6
2
3
4
5
2
3
6
8
2
3
10
14
4
6
4
10
4
18
Reflecting on Table Patterns
 Compare your patterns. How are they alike? How
are they different?
 What if we created a new table to help us with our
comparison?
 Why is there always a +2 in the pattern?
Concluding the Table Pattern
Number of sides on a
single table (s)
3
4
6
Number of People (p)
where n is the number
of tables
n+2
2n + 2
4n + 2
Correspondence for number of sides and number of
people
P = (s– 2)n +2
Task: Building with Toothpicks
1.
2.
3.
4.
5.
6.
7.
Build each shape listed on the handout. Use a ‘T-chart’ (table) to record the
perimeter of each shape.
Next, predict the perimeter of the 5th shape in the sequence. Explain your
prediction. Build the shape to verify your prediction.
What stays the same (constant) moving from shape to shape (or, ‘stage to
stage,’ or ‘term to term’)? What changes (variable)?
Write at least one rule to find the perimeter for any shape n and justify your
reasoning.
What is the perimeter for shape 30?
How did you use additive thinking? Or, did you use multiplicative thinking?
Both? Describe in words your thinking.
Rebuild the shapes making only the perimeter with toothpicks. What
changes? What stays the same? Describe any new insights for the problem.
---Friel, et. al., 2002, p. 75
Mathematize the ‘Building with Toothpicks’
problem
 What are the ‘big mathematics ideas’ of this problem?
 Identify different ‘models’ that can be used to solve this
problem.
 What ‘strategies’ may be used to solve this problem?
 What specific VA Standards of Learning are connected to this
problem? Make a list.
 Be ready to share in whole-group discussion.
Teaching Task: Analyzing Student Work
Examine each of the students’ responses.
Describe the thinking of each student …

What relationships did each student find?


Are they correct? Why or Why not?
Any surprises?
As a teacher, what would you do with each
student to follow-up on his/her specific
work?
---Friel, et. al., 2002, pp. 14-16
Our Focus: 5 Big Ideas

The properties of arithmetic used with the four operations are the same
for algebra.

An equation uses an ‘equal sign’ to show that two quantities are
equivalent.

A variable is a tool that describes relationships in many different ways.

Quantitative reasoning extends relationships between and among
quantities.

Functional relationships are expressed in words, symbols, tables, or
graphs, and reasoning occurs between these representations.

1.
---Blanton, et al., 2011, pp. 12-14.
What is mathematics?
“Mathematics is a way of thinking that
involves studying patterns, making
conjectures, looking for underlying structure
and regularity, identifying and describing
relationships, and developing mathematical
arguments to show when and why these
relationships hold.”
---Russell,
Schifter, & Bastable, (2011), p. 2.
Classroom Talk…
Promotes Student Understanding
Classroom dialogue may provide direct access to ideas,
relationships among those ideas, strategies, procedures,
facts, mathematical history, and more…
Classroom dialogue also supports student learning
indirectly, through the building of a social
environment—a community—that encourages learning…
students are encouraged to treat one another as equal
partners in thinking, conjecturing, exploring, and
sharing ideas.
---Chapin, S.H., O’Connor, C., & Anderson, N.C. (2009). Classroom
Discussions: Using Math Talk To Help Students Learn, Grades 1–6, p. 6.
Five Productive Teacher Talk Moves
1.
Revoicing: To help with the lack of clarity in a
student’s verbalization of his/her ideas; e.g., “So
you’re saying that it’s an odd number?”
2.
Repeating: Asking students to restate someone
else’s reasoning; e.g., “Can you repeat what he just
said in your own words?”
3.
Reasoning: Asking students to apply their own
reasoning to someone else’s reasoning; e.g., “Do
you agree or disagree, and why?”
Five Productive Teacher Talk Moves
4.
Adding On: Prompting students for further
participation; e.g., “Would someone like to add
something more to this?” or, Who would like to
add something more to that response?
5.
Waiting: Using wait time; e.g., “Take your
time… we’ll wait…”
---Chapin, S.H., O’Connor, C., & Anderson, N.C.
(2009). Classroom Discussions: Using Math Talk
To Help Students Learn, Grades 1–6, p. 13.
Three Productive Talk Formats
 Whole-Class Discussion
 Small-Group Discussion
 Partner Talk
---Chapin,
S.H., O’Connor, C., & Anderson, N.C. (2009).
Classroom Discussions: Using Math Talk To Help Students
Learn, Grades 1–6, p. 18.
Two Types of Division
24 ÷ 8
Partitive (Sharing):
Put 24 things into
8 groups; how many in each
group?
24 ÷ 8
Measurement or Quotative
(Grouping): Make groups of 8;
how many groups are there?
Partitive Division:
Sharing 24 things among 8 people
24 ÷ 8
The divisor is the number of groups. How
many in each group?
Story context:
I made 24 cookies to share among 8
people. How many cookies will each
person receive?
Measurement Division:
Making groups of 8
24 ÷ 8
The divisor is the number in each group.
How many groups?
Story context:
I have 24 cookies to put in bags of 8.
How many bags will I fill?
Create a story context that works with all of these
expressions. Can you develop
both types of division situations?
24 ÷ 8
24 ÷ 3
24 ÷ 1/2
24 ÷ 1/4