Geoffrey Shen is a mathematics teacher in SUIS Qingpu’s High School. He graduated from the Department of Mathematics of Fudan University with a bachelor’s degree and received a doctorate from the City University of Hong Kong. He has actively engaged in the teaching of mathematics in international education contexts for many years. Having accumulated much experience over those years, he is particularly good at explaining the most complex mathematics in the simplest language.
We invited Geoffrey Shen to lead us through an exercise, starting from the seemingly simplest “Why does 3 times 4 equal 12?”, triggering and expanding our in-depth thinking on mathematical problems, and cultivating core mathematical literacy.
One morning, when our family was having breakfast together, I asked my son:
“What is 3 times 4?”
“12!” my son answered immediately.
“Yes. So why does 3 times 4 equal 12?”
“Because 3 candies were given to 4 children…”
…
I had recently taught him the concept of division. Now, it seemed that he was confusing the concepts of multiplication and division.
“Think about this question carefully today. If you can’t figure it out, you can ask your teacher.”
That night, when we had dinner together, I asked him again: “Why does 3 times 4 equal 12?”
“Because three 4’s add up to 12.”
This time, he got the answer right.
“Did you ask your teacher?”
“I remembered it myself. The teacher taught us in our lesson.”
Why does 3 times 4 equal 12?
Because the sum of three 4s, or the sum of four 3s, equals 12.
4 + 4 + 4
= 3 + 3 + 3 + 3
= 12。
If we ask another question, “Why does 3.3 times 4.4 equal 14.52?”, I guess that not many people would be able to give a right explanation.
The multiplication of two decimals or fractions obviously cannot be a simple repeated addition like the multiplication of two positive integers. So how can we explain it?
For example, 3 times 4.4 is the sum of three 4.4’s.
So, 3 times 4.4 equals 13.2.
Multiplying a positive integer by a decimal is as easy to understand as the multiplication of two positive integers.
If we apply the same concept, multiplying 3.3 by 4.4 is 3.3 4.4’s added together.
Therefore, the key to the question is, how to understand 3.3?
It should be noted that the Arabic numerals we most commonly use today are a very concise and efficient way to express quantities.
For example, the number 3333 has four 3’s in it. They look the same, but the meaning of each 3 is quite different.
The first 3 means three 1000’s; the second 3 means three 100’s; the third 3 means three 10’s; and the fourth 3 means three 1’s.
The same number appearing in different positions have different meanings – this is definitely a brilliant idea that only the smartest people can think of!
What is 0.1?
In the decimal system, 10 is 10 ones, and 0.1 is one-tenth of 1.
After thoroughly understanding the expression of Arabic numerals, you will find it easy to understand what 3.3 times 4.4 is.
3.3 is three 1’s, plus three 0.1’s, which is three 1’s, plus one-tenth of three 1’s.
So, 3.3 times 4.4 is 3 times 4.4, plus one-tenth of 3 times 4.4.
= 3 x 4.4 + 0.3 x 4.4
= 3 x 4.4 + 3 x 4.4 / 10
= 13.2 + 1.32
= 14.52
Although the above deduction seems a bit complicated, the general idea is consistent with that of explaining why 3 times 4 equals 12.
After understanding why 3.3 times 4.4 is equal to 14.52, are you beginning to believe that multiplication is nothing but ‘a piece of cake’?
You know that -3 times -4 equals 12. But do you know why? 🙂
