Eshwarappa, a farmer in Chintamani, Karnataka, visits the market for rice seeds. He overhears a conversation about the decline in rice varieties. Ramanna states that earlier there were about one lakh varieties of rice, but now only a handful remain. Lakshmamma mentions a seed bank near her house with about a hundred indigenous varieties.
The concept of a lakh is introduced. One lakh is $1,00,000$.
Estu, Eshwarappa's son, is surprised by the number of rice varieties, wondering if one could taste all of them in a lifetime.
The chapter explores the magnitude of a lakh by observing number patterns:
Roxie and Estu discuss how many rice varieties could be tasted in a lifetime. If one variety is tasted per day, it's far from a lakh in 100 years ($365 \times 100 = 36,500$). They consider if tasting 2 or 3 varieties a day would make a difference.
Large numbers are often better understood by comparison to familiar objects or quantities.
An example is given with Somu, who is $1$ metre tall. If each floor of a building is about four times his height ($4$ metres), a $10$-floor building would be $40$ metres tall.
Questions are posed to compare the heights of the Statue of Unity and the Kunchikal waterfall to Somu's building.
Roxie and Estu have different perspectives on whether a lakh is a large number:
The Indian place value system uses commas to help read large numbers. Commas are placed after the thousands place and then after every two digits (e.g., $1,00,000$ for one lakh).
Exercises are provided to write numbers in words.
This section introduces special calculators with specific buttons to understand place values and large numbers.
This calculator has only a $+1000$ button. Exercises involve determining how many times the button must be pressed to reach numbers like $3000$, $10,000$, $53,000$, $90,000$, and One Lakh.
This calculator has only a $+10$ button. Exercises involve determining how many times the button must be pressed to reach numbers like $500$, $780$, $1000$, $3700$, $10,000$, and One Lakh.
This calculator has only a $+100$ button. Exercises involve determining how many times the button must be pressed to reach numbers like $400$, $3700$, $10,000$, $53,000$, $90,000$, $97,600$, and One Lakh.
Questions also explore how many hundreds are needed to make ten thousand or one lakh.
This calculator has multiple buttons: $+1, +10, +100, +1000, +10000, +100000, +1000000$. It can achieve numbers in multiple ways. For example, to get $321$, it can press $+10$ thirty-two times and $+1$ once, or $+100$ two times, $+10$ twelve times, and $+1$ once.
Exercises involve finding different ways to obtain numbers like $5072$, $8300$, $40629$, $56354$, $66666$, and $367813$ using button clicks and writing expressions for them.
This calculator aims to use buttons as minimally as possible. It has buttons: $+1, +10, +100, +1000, +10000, +100000$.
Exercises involve finding the minimum number of button clicks to obtain numbers like $5072$ and $8300$. The smallest number of button clicks for a number corresponds to the sum of its digits in its Indian place value notation (e.g., $5072 = 5 \times 1000 + 0 \times 100 + 7 \times 10 + 2 \times 1$, so $5+0+7+2 = 14$ clicks).
The chapter introduces even larger numbers like crore and arab.
A table compares the Indian System and the American System (also called the International System) for naming numerals and placing commas:
The Indian system is followed in several South Asian countries. The words lakh and crore originate from Sanskrit words lakṣha (लक्ष) and koṭi (कोटि).
Key relationships:
Exercises involve reading and writing numbers in both systems and comparing them.
Often, exact numbers are not required, and an approximation is sufficient to give an idea of quantity.
There are situations where it's better to round up (approximated number is more than actual), like ordering $750$ sweets for $732$ people.
There are situations where it's better to round down (approximated number is less than actual), like saying an item costing $₹470$ is around $₹450$.
Students are encouraged to think about situations where rounding up, rounding down, either is okay, or exact numbers are needed.
For large numbers, it's useful to know their nearest thousand, lakh, or crore. This involves rounding numbers to different place values.
Example: The nearest neighbours of $6,72,85,183$ are:
Exercises involve finding nearest neighbours for given numbers and estimating sums and differences, then evaluating the accuracy of these estimates.
A table provides the populations of various Indian cities in $2011$ and $2001$. Questions based on this data require students to make observations and calculations by approximation, such as population increase and cities whose populations have almost doubled.
This section explores multiplication shortcuts and patterns.
Roxie's method for $116 \times 5$: $116 \times 5 = 116 \times \frac{10}{2} = 58 \times 10 = 580$.
Estu's method for $824 \times 25$: $824 \times 25 = 824 \times \frac{100}{4} = 206 \times 100 = 20600$.
The concept is to multiply by powers of $10$ and then divide, simplifying calculations.
Exercises involve finding quick ways to calculate products like $2 \times 1768 \times 50$ and $72 \times 125$.
This sub-section presents multiplication problems that produce interesting patterns, such as $11 \times 11 = 121$, $111 \times 111 = 12321$, etc. Students are asked to evaluate these and extend the patterns.
Questions also explore the number of digits in products of numbers with a certain number of digits (e.g., two 2-digit numbers can result in a 3- or 4-digit number).
This section presents interesting facts involving large numbers, requiring students to perform calculations (multiplication or division) to uncover the facts and then read the numbers in both Indian and American naming systems.
This section encourages students to think critically about large numbers and make estimations based on real-world scenarios.
This section presents puzzles involving forming numbers using lines (like toothpicks or matchsticks) to represent digits. Students are asked to form numbers, add or rearrange sticks to make bigger numbers, and find the largest/smallest numbers possible with a given number of sticks.