Class 6 Maths Chapter 3 Notes Number Play
Class 6 Maths Notes Chapter 3 – Class 6 Number Play Notes
→ Numbers can be used for many different purposes, including to convey information, make and discover patterns, estimate magnitudes, pose and solve puzzles, and play and win games.
→ Thinking about and formulating set procedures to use numbers for these purposes is a useful skill and capacity (called “computational thinking”).
→ Many problems about numbers can be very easy to pose, but very difficult to solve. Indeed, numerous such problems are still unsolved (e.g., Collatz’s Conjecture).
Numbers are used in different contexts and in many different ways to organize our lives. We have used numbers to count, and have applied the basic operations of addition, subtraction, multiplication, and division on them, to solve problems related to our daily lives. In this chapter, we will continue this journey, by playing with numbers, seeing numbers around us, noticing patterns, and learning to use numbers and operations in new ways.
Numbers can Tell us Things Class 6 Notes
What are these numbers telling us?
Some children in a park are standing in a line. Each one says a number.
What do you think these numbers mean?
The children now rearrange themselves, and again each one says a number based on the arrangement.
A child says ‘1’ if there is only one taller child standing next to them.
A child says ‘2’ if both the children standing next to them are taller. A
child says ‘0’, if neither of the children standing next to them are taller.
That is each person says the number of taller neighbours they have.
Supercells Class 6 Notes
Observe the numbers written in the table below. Why are some numbers coloured? Discuss.
A cell is coloured if the number in it is larger than its adjacent cells. 626 is coloured as it is larger than 577 and 345 whereas 200 is not coloured as it is smaller than 577. The number 198 is coloured as it has only one adjacent cell with 109 in it, and 198 is larger than 109.
Let’s do the supercells activity with more rows. Here the neighbouring cells are those that are immediately to the left, right, top, and bottom.
The rule remains the same: a cell becomes a supercell if the number in it is greater than all the numbers in its neighbouring cells.
In Table 1, 8632 is greater than all its neighbours 4580, 8280, 4795, and 1944.
Complete Table 2 with 5-digit numbers whose digits are ‘1’, ‘0’, ‘6’, ‘3’, and ‘9’ in some order. Only a coloured cell should have a number greater than all its neighbours.
The biggest number in the table is 96,301.
The smallest even number in the table is 19,306.
The smallest number greater than 50,000 in the table is 60,193.
Once you have filled the table above, put commas appropriately after the thousands digit.
Patterns of Numbers on the Number Line Class 6 Notes
We are quite familiar with number lines now. Let’s see if we can place some numbers in their appropriate positions on the number line. Here are the numbers: 2180, 2754, 1500, 3600, 9950, 9590, 1050, 3050, 5030, 5300 and 8400.
Playing with Digits Class 6 Notes
We start writing numbers from 1, 2, 3 … and so on. There are nine 1-digit numbers. Find out how many numbers have two digits, three digits, four digits, and five digits:
Digit Sums of Numbers
Komal observes that when she adds up digits of certain numbers the sum is the same. For example, adding the digits of the number 68 will be the same as adding the digits of 176 or 545.
Digit Detectives
After writing numbers from 1 to 100, Dinesh wondered how many times he would have written the digit ‘7’!
Pretty Palindromic Patterns Class 6 Notes
What pattern do you see in these numbers: 66, 848, 575, 797, 1111?
These numbers read the same from left to right and from right to left. Try and see. Such numbers are called palindromes or palindromic numbers. All palindromes using 1, 2, 3.
The numbers 121, 313, and 222 are some examples of palindromes using the digits ‘1’, ‘2’, ‘3’.
Reverse-and-add palindromes
Now look at these additions. Try to figure out what is happening.
Steps to follow: Start with a 2-digit number. Add this number to its reverse. Stop if you get a palindrome; otherwise, repeat the steps of reversing the digits and adding. Try the same procedure for some other numbers, and perform the same steps. Stop if you get a palindrome. There are numbers for which you have to repeat this a large number of times.
The Magic Number of Kaprekar Class 6 Notes
D.R. Kaprekar was a mathematics teacher in a government school in Devlali, Maharashtra. He liked playing with numbers very much and found many beautiful patterns in numbers that were previously unknown. In 1949, he discovered a fascinating and magical phenomenon when playing with 4-digit numbers.
Follow these steps and experience the magic for yourselves! Pick any 4-digit number, say 6382.
Take different 4-digit numbers and try carrying out these steps. Find out what happens. Check with your friends what they got.
You will always reach the magic number ‘6174’! The number ‘6174’ is now called the ‘Kaprekar constant’.
Carry out these same steps with a few 3-digit numbers. What number will start repeating?
Clock and Calendar Numbers Class 6 Notes
On the usual 12-hour clock, there are timings with different patterns.
For example, 4:44, 10:10, 12:21.
Try and find out all possible times on a 12-hour clock for each of these types.
Manish has his birthday on 20/12/2012 where the digits ‘2’, ‘0’, ‘1’, and ‘2’ repeat in that order.
Find some other dates of this form from the past.
His sister Meghana had her birthday on 11/02/2011 where the digits read the same from left to right and from right to left.
Find all possible dates of this form from the past.
Jeevan was looking at this year’s calendar. He started wondering, “Why should we change the calendar every year? Can we not reuse a calendar?” What do you think?
You might have noticed that last year’s calendar was different from this year’s. Also, next year’s calendar is different from the previous years. But, will any year’s calendar repeat again after some years? Will all dates and days in a year match exactly with that of another year?
Mental Math Class 6 Notes
Observe the figure below. What can you say about the numbers and the lines drawn?
Numbers in the middle column are added in different ways to get the numbers on the sides (1500 + 1500 + 400 = 3400). The numbers in the middle can be used as many times as needed to get the desired sum. Draw arrows from the middle to the numbers on the sides to obtain the desired sums. Two examples are given. It is simpler to do it mentally!
38,800 = 25,000 + 400 × 2 + 13,000
3400 = 1500 + 1500 + 400
Adding and Subtracting
Here, using the numbers in the boxes, we are allowed to use both addition and subtraction to get the required number. An example is shown.
An example of adding two 5-digit numbers to get another 5-digit number is 12,350 + 24,545 = 36,895.
An example of subtracting two 5-digit numbers to get another 5-digit number is 48,952 – 24,547 = 24,405.
Playing with Number Patterns Class 6 Notes
Here are some numbers arranged in some patterns. Find out the sum of the numbers in each of the below figures. Should we add them one by one or can we use a quicker way? Share and discuss in class the different methods each of you used to solve these questions.
An Unsolved Mystery – the Collatz Conjecture! Class 6 Notes
Look at the sequences below—the same rule is applied in all the sequences:
a. 12, 6, 3, 10, 5, 16, 8, 4, 2, 1
b. 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1
c. 21, 64, 32, 16, 8, 4, 2, 1
d. 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1
Do you see how these sequences were formed?
The rule is: one starts with any number; if the number is even, take half of it; if the number is odd, multiply it by 3 and add 1; repeat.
Notice that all four sequences above eventually reached the number 1. In 1937, the German mathematician Lothar Collatz conjectured that the sequence will always reach 1, regardless of the whole number you start with. Even today—despite many mathematicians working on it — it remains an unsolved problem as to whether Collatz’s conjecture is true! Collatz’s conjecture is one of the most famous unsolved problems in mathematics.
Make some more Collatz sequences like those above, starting with your favourite whole numbers. Do you always reach 1?
Do you believe the conjecture of Collatz that all such sequences will eventually reach 1? Why or why not?
Simple Estimation Class 6 Notes
At times, we may not know or need an exact count of things and an estimate is sufficient for the purpose at hand. For example, your school headmaster might know the exact number of students enrolled in your school, but you may only know an estimated count. How many students are in your school? About 150? 400? A thousand?
Paromita’s class section has 32 children. The other 2 sections of her class have 29 and 35 children. So, she estimated the number of children in her class to be about 100. Along with Class 6, her school also has Classes 7–10 and each class has 3 sections each. She assumed a similar number in each class and estimated the number of students in her school to be around 500.
Games and Winning Strategies Class 6 Notes
Numbers can also be used to play games and develop winning strategies. Here is a famous game called 21. Play it with a classmate. Then try it at home with your family!
Rules for Game #1:
The first player says 1, 2, or 3. Then the two players take turns adding 1, 2, or 3 to the previous number said. The first player to reach 21 wins!
Play this game several times with your classmates. Are you starting to see the winning strategy?
Which player can always win if they play correctly? What is the pattern of numbers that the winning player should say?
There are many variations of this game. Here is another common variation:
Rules for Game #2:
The first player says a number between 1 and 10. Then the two players take turns adding a number between 1 and 10 to the previous number said. The first player to reach 99 wins!
Play this game several times with your classmates. See if you can figure out the corresponding winning strategy in this case! Which player can always win? What is the pattern of numbers that the winning player should say this time?
Make your variations of this game decide how much one can add at each turn, and what number is the winning number. Then play your game several times, and figure out the winning strategy and which player can always win!
