Dare to be unconventional!
Monica Kochar
A classroom scene
“Multiply 244 by 12”, the teacher said.
And the students went fast, multiplying the following way.
2 4 4
X 1 2
————-
4 8 8
2 4 4 0
————-
2 9 2 8
Excitedly some students shouted, “Answer is 2928 Ma’am!”
“Excellent”, said the teacher!
A slow voice among the students asked, “Why do we write a 0. Why do we shift numbers”?
The teacher said, thinking of ‘syllabus’, “It is a rule. Just do it.”
And the teacher just became a ‘math murderer instead of a math creator”.
Are you a math murderer or creator?
Math murderers stifle curiosity in the students. Math creators not only satisfy it, but also show math as a creative subject. Which one are you?
I am reminded of a teacher at one of my workshops who said at the end of the workshop, “Just today I have discovered that I am a math murderer and not a creator. Give me at least three days to digest this and move on!”
Coming back to our question:
2 4 4
X 1 2
———
4 8 8
2 4 4 0
——–
2 9 2 8
——–
Where many books also write an ‘x’ to signify a shift in the digits to the left…just why do we do it?
2 4 4
X 1 2
——
4 8 8
2 4 4 x
——-
2 9 2 8
——–
The answer is in place value. Let us rewrite the problem with the place value chart.
Thousands Hundreds Tens Units
2 4 4
X 1 2
—————————
4 8 8
2 4 4 0
—————————
2 9 2 8
—————————-
Or, for short, we write it as:
Th H T U
2 4 4
X 1 2
————
4 8 8
2 4 4 0
————
2 9 2 8
————-
Let us focus on the 1st digit only, that is ‘2’. It is in the Units place.
2 Units x 4 Units = 8 Units
2 Units x 4 Tens or 40 Units = 80 Units or 8 Tens
2 Units x 2 Hundreds or 200 Units = 400 Units or 4 Hundreds
Hence our answer is 4H and 8T and 8U or 488
Th H T U
2 4 4
X 1 2
————-
4 8 8
Now let’s focus on the 2nd digit, 1. Note that it is in the Tens place, hence we are multiplying by 1 Ten.
Th H T U
2 4 4
X 1 2
————-
2 4 4
—————
Hence the complete multiplication is:
Th H T U
2 4 4
X 1 2
—————-
4 8 8
2 4 4 0
—————–
2 9 2 8
—————-
The logic should be clear I suppose by now!
Do I have to multiply from right to left always?
That is, do I have to always start from the Units place?
The starting point for multiplication in this problem is “2”.
Th H T U
2 4 4
X 1 2
——————-
4 8 8
2 4 4 0
——————
2 9 2 8
—————–
But…but…but…is it necessary? What if I start multiplying from the Tens digit?
I will multiply from Left to Right
All I have to remember is that I am multiplying by a
Ten!
Th H T U
2 4 4
X 1 2
1Ten x 4Units = 10Units x 4Units = 40 Units or
4Tens.
Hence I write 4 in the Tens place.
1Ten x 4Tens = 4Hundreds.
Hence I write 4 in H column
1T x 2H = 10U x 200U = 2000U or 2Thousands
Hence 2 goes in the Th column
There is nothing for the Units column, hence I write an x or a 0!
Th H T U
2 4 4
X 1 2
————-
2 4 4 0
————-
The 2nd row is created by using the multiplier ‘2’
2U x 4U = 8Units
2U x 4T = 8Tens
2U x 2H = 4H
Th H T U
2 4 4
X 2
————–
4 8 8
—————-
Hence our answer is, once again, 2928!
Th H T U
2 4 4
X 1 2
—————
2 4 4 0
+ 4 8 8
—————–
2 9 2 8
—————-
It’s just a ritual to multiply by the digit at the extreme right. This ritual was created by a human mind. We must challenge it for the students to make them more resilient and therefore creative in their thinking.
Try one for yourself
Simple problem – multiply 768 by 34.
For the first time, start from the number on the right, 4, and continue. This is the ‘usual way’ of working
Th H T U
7 6 8
X 3 4
—————-
+
Answer————
———————
For the second time, start from the left number, 3, and continue. This is the ‘new way’ introduced.
Th H T U
7 6 8
X 3 4
—————
+
Answer ————-
———————–
Do you get the same answer?
If yes, then why do you insist on starting from the right?
Harder problem – multiply 234 by 123
Now, we have three possible starting points! Start with ‘3’, the usual way
Th H T U
2 3 4
X 1 2 3
—————–
+
Answer—————
———————
Try something different and start with ‘2’
Th H T U
2 3 4
X 1 2 3
—————-
+
Answer————-
——————–
Now try starting with ‘1’
Th H T U
2 3 4
X 1 2 3
——————-
+
Answer————-
——————-
Extension questions
a. Do you have to start with numbers in the lower row only? What if you started with the upper row?
b. Do you have to go in sequence left to right or reverse? What if you went haphazard?
Actually, so long as you are aware in which place value column to place the answer digit, you can start from anywhere, the destination is the same!
Classroom tips
1. Always use H, T, U when teaching multiplication. Encourage students to write it as shown in the article.
2. Let students talk aloud while working – “I am multiplying a Unit by a Ten …etc”.
3. For gifted students – At the end of multiplication, introduce ‘multiplication from any starting point’.
Resource
Knowing and Teaching Elementary Mathematics;
Liping Ma; Routledge 2010.
I would urge everyone to read this book. This was an eye opener as to how Math can be made more creative at the fundamental level – numbers in primary classes.
The author creates various kinds of multi-sensory learning environments for mathematics classes whereby learning the subject becomes a fun-filled experience. Her USP is her capacity to see the child’s learning perspective in any situation. Now, she uses this extremely important element while working with adults in the zones of teacher training, content development, testing, and writing. She can be reached through her website www.humanemaths.com.
