A Progression for the Development of Equations

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A Progression for the Development of Equations
Stage 2 and 3
 Story problems are told to the class. These are recorded by
the teacher as equations
For example John has six marbles and loses some. If he has
4 left, how many did he lose?
Recorded as 6 -  = 4
Problems should include all three styles – start unknown,
change unknown and result unknown
Students then solve these problems by using materials
Stage 4
 Students work with mentally solving written box problems of
the type outlined above
 Consideration of problems where the quantity is doubled,
leading to the teacher recording  +  = 6
 Students consider more general problems like  +  = 9, with
a range of possible answers
Transition to stage 5
 Teacher recording of the  +  = 6 story problem as
2=6
 Teacher records stories with multiplications by 2, 5, 10 as
box problems
 Students work with mentally solving written box problems of
the type outlined above
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 Teacher introduces number machines with a single operation.
Students use a single machine to work out what would happen
if various numbers are input.
 Students practice recording story problems involving +/- as
box equations, then solve them numerically
Transition to stage 6
 Teacher records stories with multiplications as box equations
 Students mentally solve problems written as this sort of box
equation
 Number machines to show more than one operation. Using
flags to show a number machine in operation.
 Use tables to show how a number machine can change
numbers
 Students practice recording story problems involving any
single operation as box equations, then solve them numerically
Stage 6
 Students record simple story problems (with either +/- or
/ only) as drawings, then solve them.
 Recording input-output pairs for a number machine as
ordered pairs
 Reversing a number machine by reversing the flags
 Inverse/opposite operations
Transition to stage 7
 Students mentally solve box equations like   3 + 4 = 19
(whole number solutions only)
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 Students use drawings to help solve more complex word
problems (involving two operations).
 Students convert drawings to box equations
Stage 7
 Students use drawings and/or box equations to solve word
problems with unknowns on both sides of the equals sign
 Students use flags to identify what to do first in a box
equation (The BODMAS link)
Transition to stage 8
 Replacement of the box with an x
 Students mentally solve equations like x  3 + 4 = 19 (whole
number solutions only)
 Formal solving of equations as
x+2=3
x+2–2=3–2
x =1
 Students start working with formal solving of simple
equations involving sets of numbers other than whole numbers
 Removal of the multiplication sign, and writing divisions in
fraction form in equations
Stage 8
 Students work mentally and formally with solving a range of
equation types
 Students write equations for, and solve, word problems
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The Development of Equations
Accompanying Notes
Equations start to appear very early on in the teaching of numeracy. Earliest experiences are found
within the addition/subtraction domain when moving from one-to-one counting on materials (stage
2) to one-to-one counting by imaging (stage 3). At this level, students are generally too
inexperienced to be doing their own recording of the problems, but will see their teacher recording a
problem like “Peter had 8 marbles in his pocket, but lost some. If he ended up with 5 marbles, how
many did he lose?” as 8 -  = 5. What the students are expected to do however, is solving story
problems mentally (as simple number problems), without recording them first in an algebraic form.
In this progression, three threads can be seen in the developing use of equations. The first of these
is the use of an equation to summarise (“mathematise”) a word problem. In this thread, students are
recording a word problem using the symbols and notation of the mathematical language.
Associated to this is the second thread, the task of solving the equation (to answer the problem).
Progressively more formal techniques for this need to be introduced so students have strategies for
dealing with the more sophisticated problems that are encountered in higher levels of the
curriculum. The third thread is the concept of a “number machine” (which could be talked of as a
computer). This thread links equations to the work done already on mappings and patterns, and is a
useful context for the consideration of inverse operations and processes, which are essential to the
development of formal equation solving. When x’x and y’s are added to the number machine, they
also contribute to the starting point to functions and relations, linking with the development of
ordered pairs and graphs through patterning.
Things to note in the use of equations
1)
Use of the equals sign
Within an equation, the equals sign signals that what is on one side of the equation is
equivalent to that on the other side. For example 4 + 2 = 6. This is different to the
interpretation of most students, for a problem like 4 + 2 = , leads them to develop the
understanding that = means “work out the answer”. Consequently, the concept that the
equals signs is like a balance, with equal things on either side, needs to be developed.
When writing equations, only one equals sign should be written on a line, this is to make
sure that the equations that are written down make sense.
For example:When Tonu was explaining how he got the answer to the problem 27 + 55 he
said “I first added 27 and 50 to get 77 then I added 5 to get 82”. If this is written
down as it is spoken, 27 + 50 = 77 + 5 = 82, which makes it look like 27 + 50 = 82,
which is obviously wrong. This form of mathematics writing is called “Running
arithmetic”, and should always be discouraged. Instead the problem and its solution
should be written as:
27 + 50 = 77
77 + 5 = 82
2)
Start unknown, change unknown and result unknown
From very early on, students need to be exposed to problems that have the unknown in a
variety of places.
For example
2+3=
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+5=6
2+=5
The main issue within such work is the place of writing an equation to solve. Pupils may
find it easy to solve a word problem mentally (especially if the “story” is told to them), or
even be able to solve a written equation. However, taking a word equation and writing an
appropriate sentence with the unknown in the right position may take some doing, so may
not be introduced (or mastered) until much later than actually being able to work out the
answer of a problem
3)
Development of the use of the “box”
a)
b)
c)
d)
4)
Learning Experiences to support the development of equations
(a)
(b)
5)
A “box” stands for a missing number that needs to be found, as in the examples
above
Two different shapes of “boxes” stand for two different numbers in a problem.
For example  +  = 9
In this circumstance children can explore a whole family of facts
Two boxes the same can stand for the same number
For example  +  = 6
This allows students to become multiplicative, leading to 2   = 6 and even 2 = 6
Two sided problems
For example  +  +4 =  + 12
In such problems children don’t have to know what the value of the box is to be able
to realise that they can take one away from each side will leave a much simpler
problem.
A balance scale, some counters and some (black) film canisters are effective
equipment. By putting counters into the canister, and balancing these against an
empty canister and some displayed counters the concept of the equals sign as the
balance can be addressed, and equations written for problems modelled by the
equipment.
By using the balance, the nature of an equation and its solution is easy to explore.
For example. In examining the equation  + 3 = 9, it is easy to see that if some
counters are removed from one side of the scales, then to maintain the balance, the
same number of counters need to be removed from the other side of the scales. This
can be recorded as follows:
+3=9
+3–3=9–3
=6
which is a standard way of recording the solution of equations, and can lead to
discussions of inverse operations among other things. (However, this sort of formal
recording/discussion by students should wait until stage 7, when they have a good
understanding of all the operations, and are starting to identify them as something
that have a meaning of their own.)
Number Machines
Number machines are another useful approach to the development of equations, and are an
alternative to box equations. However, their introduction should be delayed to later stages
of the framework, as their main use relies on a clear understanding of multiplication. Below
is a simplified version of a number machine. They are very useful for operating generally,
as any number can be dropped into the first box. They can also be used backwards, thus
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introducing inverse operations, which leads nicely into topics like changing the subject and
finding an inverse equation.
+ +3
3
2
Once formal algebra has been introduced, number machines are an effective way of
considering BODMAS within the algebra setting, as each equation can be made into a
number machine, which of course relies on the understanding of which piece of mathematics
to do first.
For example: 2x - 5 = 6
What is done first – multiplying by 2 or subtracting 5?
Notice: that reversing this order then provides a neat way to sort out the order in
which the equation should be undone to find out the starting number, that is
to solve the equation.
The Dime resources have some useful booklets that cover this approach
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A Progression for the Development of Equations
Stage 2 and 3
 Story problems are told to the class. These are recorded by the teacher as
equations
For example
John has six marbles and loses some. If he has 4 left, how
many did he lose?
Recorded as 6 -  = 4
Problems should include all three styles – start unknown, change unknown and
result unknown
Students then solve these problems by using materials
Stage 4
 Students work with mentally solving written box problems of the type outlined
above
 Consideration of problems where the quantity is doubled, leading to the teacher
recording  +  = 6
 Students consider more general problems like  +  = 9, with a range of possible
answers
Transition to stage 5
 Teacher recording of the  +  = 6 story problem as
2=6
 Teacher records stories with multiplications by 2, 5, 10 as box problems
 Students work with mentally solving written box problems of the type outlined
above
 Teacher introduces number machines with a single operation. Students use a
single machine to work out what would happen if various numbers are input.
 Students practice recording story problems involving +/- as box equations, then
solve them numerically
Transition to stage 6
 Teacher records stories with multiplications as box equations
 Students mentally solve problems written as this sort of box equation
 Number machines to show more than one operation. Using flags to show a number
machine in operation.
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

Use tables to show how a number machine can change numbers
Students practice recording story problems involving any single operation as box
equations, then solve them numerically
Stage 6
 Students record simple story problems (with either +/- or / only) as drawings,
then solve them.
 Recording input-output pairs for a number machine as ordered pairs
 Reversing a number machine by reversing the flags
 Inverse/opposite operations
Transition to stage 7
 Students mentally solve box equations like   3 + 4 = 19 (whole number solutions
only)
 Students use drawings to help solve more complex word problems (involving two
operations).
 Students convert drawings to box equations
Stage 7
 Students use drawings and/or box equations to solve word problems with unknowns
on both sides of the equals sign
 Students use flags to identify what to do first in a box equation (The BODMAS
link)
Transition to stage 8
 Replacement of the box with an x
 Students mentally solve equations like x  3 + 4 = 19 (whole number solutions only)
 Formal solving of equations as
x+2=3
x+2–2=3–2
x =1
 Students start working with formal solving of simple equations involving sets of
numbers other than whole numbers
 Removal of the multiplication sign, and writing divisions in fraction form in
equations
Stage 8
 Students work mentally and formally with solving a range of equation types
 Students write equations for, and solve, word problems
D:\533562673.doc 17/02/16
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