Experience Mathematics # 2 - Divisibility

Among all the students who tried to answer the questions presented last week, Anirudh Matange (Class VI, Ahlcons School) gave the best explanations. The answers are: 

Q1. The Commutative Law for multiplication. 

Q2. The Prime Numbers $(2, 3, 5, 7, 11, 13, 17, 19)$ give rise to only two rectangles: 

Q3. The largest number of rectangles arise from $12, 18,$ and $20.$ 

Q4. The rectangles with side $2$ can be formed by the Even Numbers: $2, 4, 6, \dots, 20.$ 

Q5. All the Multiples of $3$ (namely: $3, 6, \dots, 18$) can form rectangles with side $3.$ 

Here is another mathematical activity on the concept of divisibility. When a number $n$ divides a number $m$ evenly, we say that $m$ is divisible by $n$, or that $n$ is a factor of $m$. There are a number of tests that determine whether a given number is divisible by $2, 3, 5,$ or $9.$ By doing the following experiments, you can discover these tests for yourself. 

1. Consider the numbers: $2030, 4201, 89782, 129083, 124, 5435, 67656, 9087, 8888, 90919.$ For each number, you have to tell the last digit when the given number is multiplied by $2$. 

2. For each of the numbers in Activity 1, you have to tell the last digit when the given number is multiplied by $5$. 

3. Make a table with 2 columns. In the first column, place the multiples of $9$ less than $100$ ($9, 18, 27, 36,\dots , 81, 90$). In the second column, note the sum of their digits. 

4. Make a table with two columns. In the first column, put any $10$ multiples of $3$ less than $100$—such as $12, 15, 63, 51$—that are not all multiples of $9$. In the second column, note the sum of their digits. 

Now use your experiments and guess the answer of the following: 

Q1. Which of the following numbers is divisible by $2$: $100000, 1201, 2342, 9083, 2124, 21245, 1906, 6757, 1978, 9879.$ 

Q2. Which of the following numbers is divisible by $5$: $100000, 1201, 2342, 9083, 2124, 21245, 1906, 6757, 1978, 9879.$ 

Q3. What is the rule for checking whether a number is divisible by $9$? 

Q4. Which of the following numbers is divisible by $3$: $101010, 1201, 20112, 2124, 21223, 1906, 6757, 1978, 9879, 1080.$ 

Bright students should verify, and then prove that their guesses are correct. But the proofs, while easy, are not covered in school syllabi. For the moment, it is enough to know the tests of divisibility without knowing why they work.

Experience Mathematics # 1 -- Mathematical activities

Introduction

Children learn by doing. By doing these activities, they will experience interesting mathematical ideas. They will also gain experience in thinking mathematically. This will help them understand concepts easily, and better their performance in exams.
It is also very important to remember that the encouragement of parents and grandparents motivates a child a lot. Praise them when they show their intelligence by doing mathematical activities successfully. This will make the children work hard to become better at mathematics.

Why do children find mathematics difficult? The most important reason is that they find mathematics removed from their daily existence. However, it is not too difficult to give mathematical experiences to children. In this column, we will give an activity for children. It is a good idea for parents to help the child with the activity, if the child is studying in class I-V. For older children, show them this column and challenge them to explain the answers of the questions below.

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This activity requires 20 round pegs. The round pieces of a Carrom Board set will do nicely. You may also use 20 buttons of the same size. 

Pick a number—say 15. Take 15 pegs and make as many rectangles as you can out of them. Each time you are able to make a rectangle, reproduce the rectangle in a drawing book and note the dimensions. (You can use dots to make rectangular arrays.) For example, using 15 pegs, you can draw the following rectangles. 

Repeat this activity for each number from 1 to 20. Now try to answer the following questions:

1. When you rotate a rectangle by 90 degrees, you get another rectangle with the same number of dots. What is this law called? 
2. List the numbers that give rise to only 2 rectangles? 
3. Which number (or numbers) lead to the largest number of rectangles? 
4. What are the dimensions of all the rectangles with one side consisting of 2 dots? 
5. What are the dimensions of all the rectangles with one side consisting of 3 dots? 

It is not necessary that you will be able to answer all the questions. But it is important to try hard. In the next column, we will give explanations of the mathematics underlying this activity.