School children and their parents may find the “model method” for solving problem sums to be exhausting, frustrating and annoying. It’s even more exasperating when it seems so easy in the assessment books and model answers provided by teachers.
Before you pull your hair and wave the white flag, I will let you in on a dirty, little secret.
We are just showing you the final product! There are the other flaws in our models that we are not showing you when we present you the answer. So, don’t fret and don’t give up. There are many benefits to model drawing and here are some of them:
It allows students to visualise and understand the problem, instead of blindly following rule-based algebraic steps. This encourages thinking and problem solving skills.
The model method is a very important primer for kids to get acquainted with the abstraction of problems and more difficult maths later in their schooling.
Using this method, pupils are able to solve questions that previously would have required algebra and simultaneous equations.
Now let me show you the” true picture” of model drawing.
A.The messiness of model drawing
Model solutions presented in assessment books don’t convey the difficulty in drawing an accurate model. They don’t present the messy work usually involved. Showing the entire process is painful and time-consuming. Instead of showing simply one diagram, a writer or teacher has to present many diagrams in a step by step manner which gets tedious after many problems. Showing a sketch where the models are resized or redrafted may also imply the teacher having insufficient experience and does not instil a great deal of confidence.
Model drawing should be viewed as a problem solving skill – a thinking aid like mindmaps or pictorial sketches.
The process is to:
understand the problem and to put it into diagrams
draw meaningful relationships between the known and unknown quantities
solve for the unknowns
We should encourage students to draw and sketch and let them know that it’s ok not to get the model right at the first go.
B.The inaccuracy of model drawing
An accurate diagram does not need to be scaled, but we should be able to deduce the relationships between the unknown and known quantities.
Without knowing the actual value of the unknown, it is difficult to draw an accurate model. Yet we often see pictorially accurate model solutions from teachers and assessment books. Writers know the answer to the question posed, so they often work backwards from a refined model solution. This does not reflect the real process of model drawing and may hinder a student’s understanding.
It is not always necessary to draw an accurate representation, but there are times when it is required in order to deduce the solution.
Sue had $80 more than her brother. After Sue spent $25 of her money and her brother spent 75% of his money, they still had $420 left altogether. How much did Sue have left in the end?
The model I’ve drawn looks as follows:
Note that 5 red boxes + 55 = 420. So 1 red box = 73. But notice also that the red box, to be accurate, should be longer than the “55” box. This does not affect the answer in this case, but proportionality does matter for certain questions. It is also easy to see how some students may be confused as to why the red box is smaller than the “55” box!
Amy is one quarter as old as Betty. In 4 years time, Amy will be one-third as old as Betty. How old is Amy now?
We start off the model drawing by drawing the present:
In four years time, we know that Amy is one-third as old as Betty:
From the present B plus 4 years:
We notice that the two figures above are equal so
And so that 1 blue box = 8.
Note that in the diagrams, these two are expected to be equal, although they happen to be drawn as different lengths:
This is what often happens when solving a problem through model drawing. However, almost all the given solutions in assessment books will present the following:
Assessment books hardly point out this difficulty, and make it seem like students are required to draw accurate models like these
C.The differences from the model answer
Maths problems usually have several different solution methods. Assessment books typically present only one.
It is not uncommon to see model drawing solutions that differ from the given solution. Parents, teachers and tutors could help pupils come up with these and be prepared to accept solutions that are different from the one model solution presented in the book. Likewise, they could also help with demonstrating how a particular model may be used to solve different types of problems.
Parents who did not learn the model method or have forgotten about it often find it difficult to coach their children. They may turn to tutors for help, who in turn rely on assessment books and their model solutions.
Model drawing is messy, is typically inaccurate, and may result in a different solution method from the given answer. So if you encounter complications in your model drawing, it’s completely normal. It probably means you are on the right track.