**Question**

**Introduction**

This primary
(elementary) school mathematics examination question created quite a stir among
some parent support groups on Facebook. The
issue is that the height of the triangle seems to have been omitted.

**Pythagoras’ Theorem**

Some participants
who know secondary school mathematics were quick to suggest the use of Pythagoras’
Theorem to find the height of the triangle, which works out to be 12 cm.
This leads to the answer (2) 114 cm

^{2}, which is correct. The problem is that pupils are not taught Pythagoras’ Theorem until secondary school, and so it would seem an unfair test for the pupils. So the discussion turned to thinking of various methods by which a primary school pupil may uncover the answer without resorting to advanced knowledge.**Elimination and Educated Guessing**

Mr Teo Kai
Meng, a tutor who regularly participates in the support groups, offered some
insightful observations. Assuming that the height measurement
is a whole number of centimetres, only options (2) and (4) need to be
considered as they were divisible by 19,
which the area had to be under the said assumption. [ Another tutor, Melissa Song had a similar
idea by observing that since
the triangles D

*KLN*and D*KMN*have the**, the ratio of their areas is the ratio of their bases***same height**LN*:*MN*= 19 : 16. ] We can ignore choices (1) and (3). As we know, area = ½ ´ base ´ height . Since teachers like to catch students for being careless in forgetting to multiply by ½ (or dividing by 2), it is quite likely that option (4) was set up as a booby trap. Thus one may intelligently surmise that option (2) should be the answer.
Mr K L
Chua, a tutor who calls himself “Mathematics Specialist”, used a similar
reasoning. He worked backwards from each
of the four choices to get the heights and chose the most plausible answer. Of the two whole-number answers, (4) was
eliminated and (2) was chosen since from the diagram the height should roughly
be near to 16 cm even though the diagram was not drawn to
scale.

**Scale Drawing**

Another
participant suggested doing a scale drawing to estimate the height. Indeed this can be done, and is a good tactic
too, since this could be done quickly with a ruler and pencil.

**A Visual Solution**

Assuming
no knowledge of Pythagoras’ Theorem, it is possible to construct a visual
solution. First we take four copies (indicated
in orange/light-orange) of the right-angled triangle D

*LNK*and arrange them to surround a square of side 37 cm (indicated in green), which is the same as the longest side (hypotenuse) of the said triangle. This green area is 1 369 cm^{2}. Now we rearrange the triangular pieces as indicated by the red arrows. The areas of the orangey triangles do not change when you shift them. Neither will the green area change, since the total area everything in the containing square (orangey plus green areas) remains the same.
With pairs of right-angled triangles joined
together along their longest sides, we now obtain two green squares, the larger
of which has side 35 cm. This gives an area of 1 225 cm

^{2}. The total area of the two green squares is the same as the area of the large green sqaure before the shifting, namely 1 369 cm^{2}. Hence the area of the small square is (1 369 – 1 225) cm^{2}= 144 cm^{2}.
From here we quickly deduce that the unknown height
is 12 cm, 12 being the square root of 144.
Hence we conclude that the area of D

*KLM*is 114 cm^{2}.**Remarks**

I have
shown that it is theoretically possible for primary school pupils without knowledge
of Pythagoras’ Theorem to derive the answer in an exact manner. By the way, the method of shifting triangular
pieces as indicated above can be generalised to give a proof of Pythagoras’
Theorem.

Some
parents expressed fear that this is another one of those Cheryl-like or olympiad type of problems. Is there a
conspiracy by the school teachers to purposely set difficult questions and make
life difficult for pupils, disadvantaging those who cannot afford private
tutors? In this case, could this just
have been an

*oversight*on the part of the teacher who set the question?
Entrepreneur
John Low Jiayong and tutor John Lim sourced for and managed to obtain faithful copies
of the original question. It turned out
that some school-paper vendors had inadvertently erased the 12 cm measurement. Thus in the original question, the height

*KN*was given as 12 cm, and the measurement of 37 cm for the hypothenuse*LK*was purposely given as extraneous information to distract students.
Some
parents observed that this is after all just a multiple-choice question that carries
a credit of only one mark. If this were an exam situation
with the 12 cm omitted, it would be best
to either sacrifice the 1 mark and move on, or to use tactics like
elimination, educated guessing or estimation with scale drawing.

Anyway, it had been quite a fruitful discussion, as adults (parents and tutors) attacked this problem from many angles in a purposeful way. Many parties put in concerted effort and contributed in an engaged matter. It would be good if this type of rich discussion were enacted in our classrooms everyday among pupils and teachers, perhaps enabled by social technology. For then, pupils would deeply learn and perhaps would not be so stressed out, nor be needing so much extra external help. Private tutors could then move on to focus on value-added

Anyway, it had been quite a fruitful discussion, as adults (parents and tutors) attacked this problem from many angles in a purposeful way. Many parties put in concerted effort and contributed in an engaged matter. It would be good if this type of rich discussion were enacted in our classrooms everyday among pupils and teachers, perhaps enabled by social technology. For then, pupils would deeply learn and perhaps would not be so stressed out, nor be needing so much extra external help. Private tutors could then move on to focus on value-added

*mentorship*for 21^{st}Century Learning, instead picking up the tab where school teachers have left off.
H01. Act it out (e.g. scale
drawing)

(as a class learning activity, pupils
can use scissors to cut out paper triangles

and physically move them around)H02. Use a diagram / model

H04. Look for pattern(s)

H05. Work backwards

H07. Use guess and check

H08. Make suppositions

H09. Restate the problem in
another way

**Suitable Levels**

*****Primary School Mathematics

*****Lower Secondary Mathematics

* any precocious / gifted pupil who wants to learn

* any person of any age interested in creative problem solving

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