Class 6 Mathematics · Chapter 3 NotesNumber Play

Numbers can convey information in clever ways. For example, children standing in a line each say a number based on how many taller neighbours they have. This shows that numbers can represent relationships, not just counts. You will explore arrangements where children say 0, 1, or 2 depending on the heights of those next to them. By trying different arrangements, you can discover which sequences are possible and which are not. This activity builds your reasoning and pattern-spotting skills.

• A child says 0 if no taller neighbour, 1 if one taller neighbour, 2 if both neighbours are taller.
• The tallest child at an end cannot say 1 or 2 because they have only one neighbour.
• Sequences like 0,1,2,1,0 are possible, but 1,1,1,1,1 is not.

In a grid of numbers, a supercell is a number that is greater than all its adjacent neighbours (left, right, top, bottom). You will identify supercells and create grids with specific numbers of supercells. The largest number in a grid is always a supercell, while the smallest never is. You can also explore the maximum number of supercells possible for a given grid size. This activity helps you compare numbers and think about relative sizes.

• A supercell is greater than all its immediate neighbours.
• The largest number in a grid is always a supercell.
• The smallest number in a grid can never be a supercell.
• For a 1×n grid, the maximum number of supercells is ceil(n/2).

Number lines help us visualise the order and spacing of numbers. You will place given numbers on a number line and identify missing numbers between marked points. This reinforces your understanding of place value and the relative size of numbers. You will also practise marking the smallest and largest numbers in a sequence.

• Numbers increase from left to right on a number line.
• The distance between consecutive markings represents a fixed interval.
• You can estimate the position of a number by comparing it with nearby markings.

Digits can be combined in many ways to form numbers. You will find how many 1-digit, 2-digit, 3-digit, 4-digit, and 5-digit numbers exist. You will also explore digit sums: the sum of the digits of a number. For example, the digit sum of 68 is 14 (6+8). You will find numbers with a given digit sum and discover patterns in digit sums of consecutive numbers. Additionally, you will count how many times a particular digit appears in a range of numbers.

• There are 9 one-digit numbers, 90 two-digit numbers, 900 three-digit numbers, etc.
• Digit sum is the sum of all digits of a number.
• The digit sum of consecutive 3-digit numbers like 123, 234, 345 follows a pattern: they are multiples of 3.

Palindromes are numbers that read the same forwards and backwards, like 66, 848, or 575. You will create 3-digit palindromes using given digits. You will also learn the reverse-and-add method: start with a number, add it to its reverse, and repeat until you get a palindrome. For many numbers, this process eventually yields a palindrome. However, for some numbers like 196, it is not known if a palindrome is ever reached.

• A palindrome reads the same from left to right and right to left.
• Examples121, 333, 4554.
• Reverse-and-add47 + 74 = 121, which is a palindrome.

D.R. Kaprekar discovered a fascinating property of 4-digit numbers. Take any 4-digit number with at least two different digits. Arrange its digits to form the largest and smallest numbers, then subtract the smaller from the larger. Repeat this process with the result. Eventually, you will always reach 6174, known as the Kaprekar constant. For 3-digit numbers, the process leads to 495. This routine is a fun way to explore number patterns.

• Kaprekar constant for 4-digit numbers6174.
• Kaprekar constant for 3-digit numbers495.
• The process involves arranging digits in descending and ascending order and subtracting.

Numbers appear on clocks and calendars in interesting patterns. You will find times on a 12-hour clock that are palindromes, like 4:44 or 12:21. You will also look for dates that form palindromes, such as 11/02/2011. Additionally, you will explore whether calendars repeat after some years. Calendars repeat after 6 years if one leap year is included, or after 5 years if two leap years are included.

• Palindromic times10:01, 11:11, 12:21, etc.
• Palindromic dates01/02/2001, 02/02/2002, etc.
• Calendars repeat after 5 or 6 years depending on leap years.

Mental math helps you calculate quickly without writing. You will practise adding and subtracting numbers using strategies like breaking numbers into parts or using known sums. Estimation is about finding approximate values when exact numbers are not needed. You will estimate distances, counts, and costs in real-life situations. These skills are useful in daily life and build number sense.

• Mental mathuse known facts and properties to simplify calculations.
• Estimationmake reasonable guesses based on context.
• Examplesestimating the number of students in a school or the distance between cities.

Number games can have winning strategies. In the game of 21, players take turns adding 1, 2, or 3 to a running total, aiming to say 21. The first player can always win by starting with 1 and then saying numbers that are multiples of 4 plus 1 (1, 5, 9, 13, 17, 21). In the game of 99, where players add 1 to 10, the winning strategy involves reaching numbers that are 11 apart. You will also explore the Collatz conjecture, an unsolved problem about sequences that eventually reach 1.

• Game of 21winning numbers are 1, 5, 9, 13, 17, 21.
• Game of 99winning numbers are 99, 88, 77, ..., 11, 0.
• Collatz conjectureif even, divide by 2; if odd, multiply by 3 and add 1; repeat. It is conjectured to always reach 1.