Using Matchbox Algebra with Students
The aim of the Matchbox Algebra applet is to build students’ intuition about
algebra by helping them understand the idea of a variable as an as-yet-unknown
number. The imagery is a matchbox containing an as-yet-unknown number of
matches.
To run the application open the Matchbox Algebra folder and double-click:
Matchbox Algebra.html.
Although the software can be used as a free-standing tool, it has been designed
for use in the following three-staged teaching context:
Stage 1
The Enactive (E) stage, where learners are given simple tasks that
require them physically to handle matchboxes and matches. For example, Anna
has a matchbox (containing some unknown number of matches) and two loose
matches, Ben has seven loose matches, and they are told that they each have
the same number of matches. The task is to guess how many matches are in
Anna’s matchbox. Tasks are made slightly more challenging – for example, Anna
has a two matchboxes (each containing the same number of matches) and three
loose matches, Ben has nine loose matches.
Stage 2
The Iconic (I) stage, where learners use the Matchbox applet.
Suggestions for teaching with the Matchbox applet are provided below.
Stage 3
The Symbolic (S) stage, where learners tackle similar problems
using conventional algebraic notation.
Overview of the Matchbox applet
(a) Tools
The application is offered in four levels of sophistication, distinguished by the
range of tools available.
(i)
No tools: you must guess how many matches are in the matchbox(es).
(ii)
Add/subtract tool: you can add or subtract a chosen number of matches or
matchboxes from each side. There is also a ‘Moves’ counter (which records the
number of operations used).
(iii)
Divide tool: as well as adding and subtracting, you can divide throughout
the equation by an integer value. This represents the full tool kit.
(iv)
Traditional ‘x’ notation: equations are represented in conventional
algebraic notation and you have access to the full tool kit. Additionally there is a
‘Review’ button (which provides a summary of everything you have done). There
is also a ‘Swap sides’ button which allows you to place the ‘x’ term on the left or
right hand side, as required.
(b) Levels
The ‘levels’ settings allow you to choose the degree of difficulty of the questions,
arranged as follows:
Level 1
one matchbox and all terms are positive;
Level 2
one matchbox and some terms are negative;
Level 3
several matchboxes, all on one side of the equals;
Level 4
several matchboxes on both sides;
Level 5
similar to Level 4 but with larger numbers.
Teaching with the Matchbox Algebra applet
1
Using the ‘No tools’ option
Aims – to help the learner:
 to see that the aim of the task
is to guess the unknown
number of matches in a
matchbox;
 to become aware of an
important general principle –
that provided you do the same
operation(s) to both sides,
then equality is maintained;
 to become aware that
provided you do the same
operation(s) to both sides,
then the number of matches in
a box is unaffected;
 to start to consider strategies
for simplifying the equation
with the aim of working out the
number of matches in a box.
Set the applet to ‘No tools’, select ‘Level 1’ and the first question is displayed.
Invite the learner to guess how many matches are in the matchbox.
Enter their answer using the drop-down menu in the lower right corner of the
screen. A correct response will cause the matchbox to open, revealing the
correct number of matches, otherwise you are asked to ‘try again’.
Using progressively more difficult examples (perhaps as far as level 2 or level 3),
invite to learner to explore possible strategies for simplifying and solving the
equations.
2
Using the ‘Add/subtract tool’ option
Aims – to help the learner to:
 become confident with using the
strategy of adding and
subtracting matchboxes and
matches as a way of simplifying
their equation;
 become aware that when they
simplify the equation to, say, 5
matchboxes = 15 matches, they
can mentally share the matches
equally amongst the 5 boxes,
giving 3 in each.
Using progressively more difficult examples (perhaps as far as level 4), provide
opportunities for them to practice simplifying and solving equations.
3
Using the ‘Divide tool’ option
Aims – to help the learner to:
 see that a completed ‘matchbox
algebra’ solution requires them
to finish with a single matchbox
on one side and some loose
matches on the other;
 become confident with using
the strategy of adding and
subtracting matchboxes and
matches and dividing through
by a constant as a way of
simplifying and solving their
equation.
Using progressively more difficult examples (as far as level 5), provide
opportunities for them to practice simplifying and solving equations. Note that
they are still restricted to non-negative integer solutions. Questions involving
negative and fractional solutions are included in the ‘Traditional ‘x’ notation’ tool.
4
Using the ‘Traditional ‘x’ notation’ tool
Aims – to help the learner to:
 transfer their equation-solving
strategies to equations
expressed in conventional
notation.
Choose a fairly simple (Level 3) problem and ask the learner to solve it
algebraically using the tool kit. If at any time they hit a problem, one click will
select the ‘divide’ tool (top left of the screen) and they will see the same problem
displayed in matchbox notation. Invite them to consider how they would tackle it
in this form and then one click returns them to the problem in conventional
notation. Be prepared to move between the two forms until the learner is
confident with the algebraic notation.
Note that the equation turns red when finally solved.
Gradually move up to questions of greater levels of difficulty.
At any time, click ‘Review’ and the learner can see, line-by-line what they have
done. This may provide a good opportunity to discuss the particular strategies
they used and which strategies are useful in general.
Note that when the ‘Traditional ‘x’ notation’ tool is selected, there is available an
additional sixth level of questions which allow negative and fractional solutions.