Unit Conversions in Primary Schools:
Good Answers, Bad Math
I. The Anomaly
Dollars to cents, Rupees to paisa, Kilograms to grams, all require conversion from the bigger Unit to the smaller one. This can be stated mathematically as $ 1 = 100 cents, so:
if $ 1 = 100 cents, then $2 = 2 x (100 cents that makes each $) = 200 cents.
In terms of conceptual theorizing, however, the operation subsumes the following:
“To convert a large Unit (like a $1) into smaller units (cents), you multiply it with the number of smaller units (e.g. 100 cents) that makes up the larger Unit ($).”
Therefore, $1 x 100 = 100 cents. But we know intuitively it can’t be true because $ 1 x 100 = $100, not 100 cents.
Likewise to convert 5 Kg into grams (knowing that 1000gm = 1 Kg) this is what we do:
5 Kg(large) x 1000 = 5000 gms(small).
But that proves to be counter-intuitive since 5 Kg x 1000 = 5000 Kgm, not 5000 gms.
In schools, teachers and textbooks teach children to multiply when converting large quantities into smaller ones. They perform the opposite function (divide) when converting small units to large units:
5000 gm (small) ÷ 1000 = 5 Kg (Large)
That too is incorrect: 5000 gm ÷ 1000 = 5 gm, not 5 Kg.
Should we assume that children know that they are dividing 5000 gms by 1000 gm, and not just 1000? Because if they do, it would be mathematically correct:
5000 gm ÷ 1000 gm = 5 Kg.
How? Why? What’s the difference?
II. Revisiting the Division Concept
To understand why, let’s first look at division as “Equal Sharing” below:
8 apples(i) ÷ 2 people(ii) = 4 apples/person(iii)
That’s easy enough: there are apples (i) and people (ii). The answer (iii)appears in the form of i / ii or number of apples per person
By comparison, Division as “Equal Grouping” is more subtle. If you have 8 apples and you pack them by putting 2 apples in each box, how many boxes would you need?
8 apples ÷ ( a group of) 2 apples = 4 groups of 2 apples
Since a group of 2 apples = 1 box, the answer is = 4 “boxes”
If you packed 4 apples per box, then
8 apples ÷ 4 apples = 2 boxes
So in “Division as Equal Grouping” you apply a “key” e.g. 1 box = 2 apples
Or 1 box = 4 apples. Which is why the answer appears as “boxes”.
In division as Equal Grouping, you divide the same units by the same units e.g. apples by apples, $ by $ and cents by cents:
$ 16(A) ÷ $ 8(B) = 2 groups of $ 8(C)
$ 16 ÷ $ 4 = 4 groups of $ 4
$ 16 ÷ $ 2 = 8 groups of $ 2
$ 16 ÷ $ 1 = 16 groups of $ 1
B represents the break-up of the Total (A) into smaller groups of $ 8, $ 4, $ 2 and $ 1. The smaller the groups (B), the larger the number of groups ( C ). What if the groups (B) were made even smaller, e.g. (a) half a dollar, then (b) a quarter dollar?
(a) $ 16 breaks up into how many half-dollars? $ 16 ÷ $ 1/2 = 32 groups of $1/2
We multiply 16 x 2 because the 2 halves in each dollar are repeated 16 times for $16
(b) $ 16 breaks up into how many quarter-dollars? $ 16 ÷ $ 1/4 = 64 groups of $ 1/4
We multiply 16 x 4 because 4 quarters in each dollar are repeated 16 times for $16.
The multiplication operation is therefore a follow-through of the original division operation.
IV Redefining a Unit Conversion?
Simply put, a Unit conversion involves the breaking up of a larger Unit into smaller equal units. The breaking-up involves division, not multiplication:
When we convert 5 cm into millimetres we use the key 1 cm = 10 mm. So we break up (or divide) each cm (larger unit) into a 10 equal parts (smaller units). Note the progression
5 cm divided into 2 parts = 5 ÷ 1/2 cm = 10 (halves))
5 cm divided into 4 parts = 5 ÷ 1/4 cm = 20 (quarters)
5 cm divided into 5 parts = 5 ÷ 1/5 cm = 25 (5ths)
5 cm divided into 10 parts = 5 ÷ 1/10 cm = 50 (10ths) = 50 Tenths
The Key tells us that 1 tenth of a cm = 1 mm. So 50 tenths = 50 mm
Therefore the Rules for Conversion can be expressed as follows:
A. Converting Large to Smaller units: Divide the larger unit by the number of smaller units contained in it. Both numbers should be expressed in terms of the Larger unit.
Example: Convert 25 meters into cm.
The Key here is: 1 meter (large) = 100 cm (small). That means: 1 cm = 1/100 m
It’s like asking: “How many cm fit into 25 m?”
25 m(Large) ÷ 1cm (small) = ? cm
25 m ÷ 1/100 m = 2 500 cm
We multiply 100 x 25 because 100 cm in each meter is repeated 25 times for 25 meters.
B. Converting Smaller Units to Larger: Multiply the smaller units by the number of smaller units contained in the larger Unit (expressed in terms of the Larger Unit).
In this example, $ 16 is shown to have 64 quarters as shown:
$ 16 ÷ $ 1/4 = 64 quarters
Let’s work backwards: Convert 64 quarters back into dollars.
$1 = 4 Quarters
1/4 x 64 Quarters = 16 quarters = 16 quarters ÷ 4 quarters($1) = $ 4
Example: Convert 2500 cm meters into meters.
The Key : 1m = 100 cm, so 1/100 of 1m = 1cm
Therefore 1/100 of 2 500 cm = 2 500 cm ÷ 100 cm(1m) = 25 Km
THE CORRECT RULE
Divide by 1/100
LARGE — — — — — — — — — — — — — — — — - → Small
Multiply by 1/100
Small — — — — — — — — — — — — — — — — — — → LARGE
As is obvious, Unit Conversions cannot be introduced before teaching Fractions.
However, the above rationale can help us correct the existing practice (of working with follow-through’s of the original operation) with a simple modification (see Part 2). That might help legitimize the introduction of Unit Conversions before teaching Fractions.
( Cont’d in Part 2)
© Shad Moarif, 2021 ALL RIGHTS RESERVED www.karismath.com
