5 Equations with Multiple Variables and Repeated Variables
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Expand discussion to include number sentences like x + y = 7
Extend range of number sentences further by introducing equations where variable is repeated and
consider mathematical thinking fostered here
 Start with problem situation that provides the context for thinking about number sentences with several
variables
 Consider some different equations with multiple variables and repeated variables
 Provides basis for algebraic notation and algebraic reasoning to enhance the study of arithmetic
Caged Mice: A Context for Introducing Multiple Variables
 Ricardo has 7 pet mice. He keeps them in two cages that are connected so that the mice can go back and
forth between the cages. One of the cages is big and the other is small. Show all the ways the seven
mice can be in the two cages.
 Generates 7 fact family
 Students need to develop a good way to record their answers for this problem type
 The number of ways is one more than the number
 By the end the week students made the conjecture that the number of possibilities is always one more
than the number of mice 
 Important features of this exercise in terms of mathematical engagement:
o Most children generate combinations without much order or thought of mathematical
properties – but can use some insight such as using the commutative property of addition
o Children wrestle with figuring out for certain if they have all of the possible solutions
o The struggle leads them to a conjecture about how solutions would be there for any number of
mice
o Finally express conjecture using algebraic notation
ABCDEFGHIJKLMNOPQRSTUVWXYZ
Teacher Commentary 5.1
 No matter how well you write these problems you will always have to explain parameter to some
students
 Kids like to think outside the box, but for this students will miss the point of the exercise if they do this
type of thinking
 Some kids will have no organization scheme at all
 T-chart can help to make sure they have the comparative pairs
 T-chart is also a way of understanding that these commuted pairs are different
 They eventually see thee are more solutions like fractions and negative numbers
 “When kids make connections, I know they really understand.”
A Related Problem with Number Sentences
 To make a similar problem, have to limit the domain to whole number answers
 x + y = 7 is different from equations looked at previously
 These types of number sentences open up a variety of options to engage in mathematical thinking
 Algebraic notation can be used to express these arithmetic ideas
 Students learn the concept around independent and dependent variables – once a value is chosen for s,
then the value for t becomes fixed and vice versa
 For p – r = 6 there are an infinite number of possible solutions where the difference is 6
Number Sentences with Variables Appearing More Than Once
 W + W = 12
 Can you put one number for the first W and a different number for the second W?
 Or do you have to put the same number in for both Ws?
 The second one is the one mathematicians have agreed to adopt – we say this is a rule by convention
 Mathematician’s Rule
o When the same variable appears more than once in a number sentence, the variable must be
replaced by the same number throughout the number sentence for each substitution
Solving Equations
 Once students know mathematician’s rule they are ready to try equations of this type for themselves
 This is not about doing ALGEBRA!!!
 This is to engage the students to think flexibly about number operations and relations
 These problems provide an opportunity to further extend understanding of relational thinking
o Comparing alternative strategies makes students think about number operations in different
ways
o Students start to see operations on numbers as more than just procedures
o May start to appreciate the power understanding concepts brings to using these operations
CD 5.1
 Susan uses trial and error
 Erika uses more flexible thinking
 Erika considers the problem as a whole BEFORE she starts to calculate
CD 5.2
 The solutions here are ones generated by real elementary students
 Provides students opportunities to explore relationships involving operations on numbers
 The goal here is NOT to teach the students algebra or efficient methods of solving equations
 Students eventually will learn more efficient methods for solving equations
 The value here is the PROCESS
ABCDEFGHIJKLMNOPQRSTUVWXYZ
Teacher Commentary 5.2
 “I didn’t even know that I didn’t understand math because I didn’t know that understanding was a part of
math.”
 Engaging kids with these problems helps teacher understanding grow too
 The teacher ALWAYS learns the most 
Number Sentences with Several Variables
 V + V + Y = 14
 Have many solutions
 Same number has to be substituted for each Y
A Different Use of Notation
 Text book authors often do not use  as a variable, i.e.:  +  = 9
 Important to point out to students when these inconsistencies appear in the text
 More appropriate to use  +  = 9 when two distinct variables are being considered
 Text book is NOT WRONG! Just not using  as a variable the way we do in algebra
A Common Misconception
 Over-generalization of the Mathematician’s Rule leads to students believe two different variables cannot
have the same value
 Students can be quite adamant about their assertions even when they are incorrect
Convenient Conventions
 Same variable must be replaced by the same number every place it appears
 Two different variables can be replaced by the same number or by different numbers