Colour by numbers…with a twist!
Kavitha Madhuri Reddy
When we learnt the divisibility rules in school it was by rote memorization. We did not know the ‘why’ behind these rules nor were we able to make any connections to the concepts we learnt. But, as I grew up I saw the importance of these connections as they allowed children to mathematically explain and reason out divisibility rules through pattern recognition. The activity shared below, I believe is a good starting point to introduce this concept; it can be further extended to explain more complex concepts like LCM and Algebra. My main objective of undertaking this activity was to help my 4th class students understand the divisibility rules of 2, 3, 5, 9 and 10 through a fun group activity. This activity was done in a class of 18 students.
Prior knowledge: Children should know skip-counting, multiples and simple division.
We began the class by playing a game called ‘dash’. This game involves skip-counting. A number is pre-decided by the class before the start of play. The rule of the game is to say ‘dash’ when a multiple of the pre-decided number comes instead of saying the number aloud. We started with an easy number ‘5’ and then continued to two other numbers ‘2’ and ‘3’.
After this, I asked them to form six groups of five members each. I gave each group a set of five 10×10 paper grids with some guidelines and probing questions as follows:
• Take one grid each and colour/shade all the numbers divisible by 2/3/5/9/10.
• Do you see a pattern among the numbers you coloured?
• If there is something common, could you share it with your group members?
Every child in a group was to choose one number from the given five numbers. This way all the five numbers are covered. Around six to eight minutes were given for this activity and I observed that most of the groups finished colouring and discussing well before time. Since there were six groups, I asked the first group to present only about the pattern that was seen in the multiples of ‘2’ to the whole class. They all confidently and with enthusiasm said that the numbers are ending with 0/2/4/6/8. Then, I gave them numbers more than ‘100’ and asked the class if these were divisible by ‘2’, if so, why or why not? This way, as a class we came to the conclusion that a number would be divisible by ‘2’ only when it ended with 0/2/4/6/8. The second group shared their findings on the number ‘5’ and then the third group for ‘10’. The first three numbers were comparatively easy as the students were able to identify the obvious patterns.
It was the fourth group’s turn to discuss the divisibility rule of ‘9’. I thought this was slightly challenging. As the group was reading the numbers aloud, I was writing down on the board simultaneously. Then I asked the students what pattern they observed, and one of the students said, ‘You see the numbers increasing on one side and decreasing on the other,’ then another said, “You see the numbers being reversed, for example, we see ‘18’ and ‘81’”; a third student said if we add any two numbers of the multiples of 9 their sum is also 9. This way, the students came out with interesting observations and the whole class agreed to and came to a conclusion that a number would be divisible by ‘9’ when the sum of its digits was divisible by 9.
After every divisibility rule, I presented numbers beyond 100 so that I could make sure they understood the concept. The fifth group was presenting on the pattern they could see for the number ‘3’. I asked them if the patterns observed for ‘9’could be applied to ‘3’ as well. They took some time and said, “Numbers are not reversed like with 9, nor are the numbers decreasing and increasing but the sum of the digits in the numbers are divisible by 3”.
This way the five groups presented one number each and the sixth group summarized all the rules with some guidance and support. I had been very skeptical with numbers ‘9’ and ‘3’. Would my students be able to come out with any thought-provoking observations? It was very interesting and pleasantly surprising to see them come up with such fascinating responses. However, this activity has certain limitations. The patterns are already formed and the children don’t have to come up with a pattern on their own. This is just one of the ways to observe and arrive at the divisibility rules and is not a proof.
After this, I did an oral exercise by writing numbers randomly on the board and asking for numbers which would be divisible/not divisible. They thoroughly enjoyed this exercise which helped them learn the divisibility rules in a mathematical way and not just rote learn. The next class involved a quick recap on the divisibility rules learnt followed by a worksheet on divisibility rules.
The author is a PYP Homeroom teacher for Grade 4 at Indus International School, Hyderabad. She has been a teacher for five years. She can be reached at kavitha.reddy@apu.edu.in.
