Expressions and Equations (6th Grade Learning Trajectory by Dr. Diana Moss)
*This document represents how students in a whole class teaching experiment thought about the
variable. It documents the kinds of student thinking that emerged along with problem types.
Changing Concept of Variable
Conditions
Focus of
Classroom
Activity
Forms of
Reasoning
Key
Mechanisms
that Shifted
Student
Thinking
Type of
Thinking
Conditions
Focus of
Classroom
Activity
1. Variable as Label
 Quantity is known
 Using a variable to keep a record of a specific quantity
 Finding the total of two groups by writing an expression or an
equation
adults + children = Total
a+c=T
 Using a sum of like and unlike terms to find the total amount
8 red M&Ms + 4 blue M&Ms
8r + 4b
 Abbreviating the name of a group with the first letter of the word
 Using a letter to represent one object or one group with the same
classification
 Asking students to share different representations of the expression
adults + kids = total
a+k=t
2 adults + 5 kids
2a + 5k
 Directing students to come up with a more efficient way to represent
the same expression
r+r+r+r+r=
2r + 3r =
5r
 Circling sign, coefficient, and variable to combine like terms
 Additive
Finding the Total
r + r = 2r
 Multiplicative
5•r
where r = 1
 Algebraic
Generalizing Arithmetic to Algebra
 Expressions and Equations
Expression as a sum of like and unlike terms
Equation as computing a total
2. Variable as Changing Quantity
 Variable represents changing values of a specific quantity
 Writing an expression that can be used to find the total cost
4c + s
where 4c is the cost of 4 packages of 6 cupcakes and s is the cost of a
single cupcake
Forms of
Reasoning

Key
Mechanisms
that Shifted
Student
Thinking
Type of
Thinking


Focus of
Classroom
Activity
Forms of
Reasoning
Key
Mechanisms
that Shifted
Student
Thinking
Type of
Thinking
Conditions
Multiplicative
4•c
where c is a changing quantity

Conditions
Symbolizing the price with a variable and acknowledging that this
price may change
c is the cost of a package of 6 cupcakes
Suggesting to students that the expression is a formula that can be
used at any store
Algebraic
Using symbols in a Meaningful Way to model a formula
 Expressions and Equations
Expression as modeling the cost of a known quantity
Equation as computing the total cost
3. Variable as Known Value
 Value for variable is given
 Known value can be substituted for the changing quantity
 Providing the price of an item at different stores and asking students to
find the total cost using the expression.
At Walmart, one package of six cupcakes is $6 and a single cupcake is $1.
At Safeway, one package of six cupcakes is $8 and a single cupcake is $2.
At Raley’s, one package of six cupcakes is $7 and a single cupcake is $1.
 Substituting a given value for the variable
 Understanding that the coefficient is multiplied by the value of the
variable
4c means 4 • c
 Developing the rule of “plug and chug” where once you plug in the
value, the variable disappears and terms can be combined
4c + s for
c = 6 and s = 1
4(6) + 1 = 24 + 1 = 25
 Multiplicative
4c means 4 • c
 Additive
After plug and chug, combine by adding or subtracting
 Algebraic
Using symbols in a Meaningful Way to evaluate an expression
 Expressions and Equations
Using an expression to substitute values for variables
An equation is the final computation of the total of the expression
4. Variable as Unknown Value
 Must solve for the unknown
Focus of
Classroom
Activity
Forms of
Reasoning
Key
Mechanisms
that Shifted
Student
Thinking


Unknown variable must be isolated
Balancing a scale to solve equations and Algebra Touch App for
isolating the variable.
t+2 = 6
 Using the area of a rectangle formula to find the width.
A = lw
if A is 18 and l is 6, find w.
 Learning distributive property where the variable is number of family
members and cost is known.
Let n be the number of family members. Every member of the family
needs to buy a backpack that costs $90 and a sleeping bag that costs $60.
1. Total Cost = n(90 + 60)
2. Total Cost = 90n + 60n
 Using opposite operations to solve an equation.
 The scale must be balanced.
 The equal sign means that both sides have the same value.
 Identifying when to use AP and when to use MP.
AP is Addition Property of Equality and MP is Multiplication Property
of Equality.




Type of
Thinking




“Making moves” to isolate the variable
t+2=6
-2 = -2
t=4
Subtracting 2 is making “one move” to isolate t.
Plug and chug can be used to check the answer to an equation
t + 2 = 6, so
t=4
Check: 4 + 2 = 6
6=6
Distributive property is necessary for simplifying an algebraic
expression into an equivalent expression
2(x – 3) = 2x – 6
In arithmetic, distributive property is not necessary, but produces
equivalent expressions
2(3 + 4) = 2(7) = 14 but this is also:
2(3 + 4) = 2(3) + 2(4) = 6 + 8 = 14
Additive
90n + 60n = 150n
Multiplicative
150n means 150 • n
Algebraic
Study of Structure where the equal sign means “the same as”
Expressions and Equations
Equation is the same quantity on both sides
Equation as balanced
Equation is seen as equivalent expressions
Conditions
Focus of
Classroom
Activity
Forms of
Reasoning
Key
Mechanisms
that Shifted
Student
Thinking
5. Independent and Dependent Variable
 A relationship exists between the independent and dependent variable
 If the input is known, the output can be found
 If the output is known, the input can be found
 Examining the formula for the perimeter of a square.
P = 4s
 Modeling a situation with a function using an arrow diagram,
algebraic function, and graph.
You are saving money in your piggy bank. You already have $20 and
you get $5 everyday for doing your chores. How much will you have in
your piggy bank after d days?
 Using a verbal phrase for the formula
 Writing the formula
y = 5d + 20
 Using an arrow diagram

Graphing the ordered pairs

Picking any number for x and plugging this into the equation gives a
point on the line.
The arrow diagram gives the points on the line.


Type of
Thinking
Functions can be represented as equations, arrow diagrams, and
graphs.
 Algebraic
Studying the patterns through the formula and the Domain and Range
 Functional
Building and generalizing the relationship between the function as a
formula, arrow diagram, and graph.
 Expressions and Equations
Equation is seen as the same on both sides and balanced
An equation relates two variables