Meaningful Use of Symbols
Two types of symbols most important to
algebra, but unfortunately not well
understood by many students:
Equal Sign (=) and Inequality Signs (˂, ≤, ˃,≥)
Variables
Teaching Student-Centered Mathematics by
Van de Walle, Bay-Williams, Lovin and Karp
1
Meaningful Use of Symbols
Border Tiles
Equal Sign and Inequality Signs
Conceptualizing the Equal and
Inequality Signs with a Balance
True/False Sentences
Relational Thinking
Teaching Student-Centered Mathematics by
Van de Walle, Bay-Williams, Lovin and Karp
2
Relational Thinking
• Takes place when a student observes and uses
numeric relationships between two sides of
the equal sign rather than actually computing
the amounts
• This type of thinking is a first step toward
generalizing the relationships found in
arithmetic to the relationships used when
variables are involved
Teaching Student-Centered Mathematics by
Van de Walle, Bay-Williams, Lovin and Karp
3
Consider two different
explanations for
placing a 3 in the box
for this open
sentence:
2.4 ÷
= 4.8 ÷ 6
1. “Since 4.8 ÷ 6 is 0.8,
then 2.4 ÷ something is
also 0.8, so that must be
3.”
2. “I noticed that 2.4 is half
of 4.8, so I need to
divide by a number half
the size of 6 in order to
maintain equivalence, so
the number is 3.”
Teaching Student-Centered Mathematics by
Van de Walle, Bay-Williams, Lovin and Karp
4
Meaningful Use of Symbols
Border Tiles
Equal Sign
Inequality Signs
Equality and Inequality Signs with a
Balance
True/False Sentences
Relational Thinking
Open Sentences
Teaching Student-Centered Mathematics by
Van de Walle, Bay-Williams, Lovin and Karp
5
Solving Equations Using a Balance
Scale
Teaching Student-Centered Mathematics by
Van de Walle, Bay-Williams, Lovin and Karp
6
The Meaning of Variables
Variables Used as Unknown
Values:
• One-Variable Situations
Teaching Student-Centered Mathematics by
Van de Walle, Bay-Williams, Lovin and Karp
7
• Gary ate 14 strawberries, and Jeremy at some,
too. The container of 25 strawberries was
empty! How many strawberries did Jeremy
eat?
How could we use what we learned about open
sentences to write a statement representing
this story problem? Can you use variables
instead of
?
Teaching Student-Centered Mathematics by
Van de Walle, Bay-Williams, Lovin and Karp
8
Two or More Variable Situations:
Systems of Equations
• Five Problems With Multiple
Variables
• What’s My Weight?
Teaching Student-Centered Mathematics by
Van de Walle, Bay-Williams, Lovin and Karp
9
The Meaning of Variables
Variables Used as Unknown Values:
• One-Variable Situations
• Two-or-More-Variables Situations
Systems of Equations
and Reflect
How could you help students bridge the
connection from these informal ways of solving
to a more formal understanding of systems of
equations?
Teaching Student-Centered Mathematics by
Van de Walle, Bay-Williams, Lovin and Karp
10
Use relational reasoning to determine which
ones of the following systems of equations can
be solved for or without using algebra.
Teaching Student-Centered Mathematics by
Van de Walle, Bay-Williams, Lovin and Karp
11
Simplifying Expressions and
Solving Equations
My Favorite No
https://www.youtube.com/watch?v=Rulmok_9HVs
Teaching Student-Centered Mathematics by
Van de Walle, Bay-Williams, Lovin and Karp
12
Variables Used as Quantities That
Vary
• a) If you have $10 to spend on $2 granola bars and $1 fruit
bars, how many ways can you spend all your money without
receiving change?
• Use the table below to explore ways to spend your money.
Number of $2
granola bars
Number of $1 fruit Total amount
bars
spent ($10)
Teaching Student-Centered Mathematics by
Van de Walle, Bay-Williams, Lovin and Karp
Equation?
13
b) You bought $1.75 pencils and $1.25 erasers from the school store,
and you spent exactly $35.00. What might you have purchased?
What equation represents this situation?
• Use the table below to explore values that
work and to look for patterns.
Number of $1.75 Number of $1.25 Total amount
pencils
erasers
spent ($35)
Teaching Student-Centered Mathematics by
Van de Walle, Bay-Williams, Lovin and Karp
Equation?
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