NUMBER PLAY3
Numbers are used in different contexts and in many different ways to organise 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.
Think about various situations where we use numbers. List five different situations in which numbers are used. See what your classmates have listed, share, and discuss.
Math Talk
3.1 Numbers can Tell us Things
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.
Did you figure out what these numbers represent?
Hint: Could their heights be playing a role?
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.
Try answering the questions below and share your reasoning.1. Can the children rearrange themselves so that the children standing at the ends say ‘2’?2. Can we arrange the children in a line so that all would say only 0s?3. Can two children standing next to each other say the same number? 4. There are 5 children in a group, all of different heights. Can they stand such that four of them say ‘1’ and the last one says ‘0’? Why or why not?5. For this group of 5 children, is the sequence 1, 1, 1, 1, 1 possible? 6. Is the sequence 0, 1, 2, 1, 0 possible? Why or why not?7. How would you rearrange the five children so that the maximum number of children say ‘2’?
Math Talk
3.2 Supercells
Observe the numbers written in the table below. Why are some numbers coloured? Discuss.
4379756310292834
200577626345790694109
A cell is coloured if the number in it is larger than its adjacent cells. The number 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.
Figure it Out
1. Colour or mark the supercells in the table below.
682867094353780370873088000558352
2. Fill the table below with only 4-digit numbers such that the supercells are exactly the coloured cells.
534612589635
3. Fill the table below such that we get as many supercells as possible. Use numbers between 100 and 1000 without repetitions.
4. Out of the 9 numbers, how many supercells are there in the table above? ___________
5. Find out how many supercells are possible for different numbers of cells.
Math Talk
Do you notice any pattern? What is the method to fill a given table to get the maximum number of supercells? Explore and share your strategy.
6. Can you fill a supercell table without repeating numbers such that there are no supercells? Why or why not?
TryThis
7. Will the cell having the largest number in a table always be a supercell? Can the cell having the smallest number in a table be a supercell? Why or why not?
8. Fill a table such that the cell having the second largest number is not a supercell.
9. Fill a table such that the cell having the second largest number is not a supercell but the second smallest number is a supercell. Is it possible?
10. Make other variations of this puzzle and challenge your classmates.
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.
Table 1
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.
2430750073509870
3115479591249230
4580863282803446
5785194458056034
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.
Table 2
96,30136,109
13,60960,31919,306
60,193
The biggest number in the table is ____________ .
10,963
The smallest even number in the table is ____________.
The smallest number greater than 50,000 in the table is ____________.
Once you have filled the table above, put commas appropriately after the thousands digit.
3.3 Patterns of Numbers on the Number Line
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.
2180
10002000
300040005000600070008000900010,000
2754
Figure it Out
Identify the numbers marked on the number lines below, and label the remaining positions.
a.
20102020
b.
99969997
15,077 15,07815,083c.
d.
86,705 87,705
Put a circle around the smallest number and a box around the largest number in each of the sequences above.
3.4 Playing with Digits
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.
1-digit numbers From 1–9
2-digit numbers3-digit numbers4-digit numbers5-digit numbers
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 same as adding the digits of 176 or 545.
Figure it Out
1. Digit sum 14a. Write other numbers whose digits add up to 14.b. What is the smallest number whose digit sum is 14?c. What is the largest 5-digit whose digit sum is 14?d. How big a number can you form having the digit sum of 14? Can you make an even bigger number?2. Find out the digit sums of all the numbers from 40 to 70. Share your observations with the class.3. Calculate the digit sums of 3-digit numbers whose digits are consecutive (for example, 345). Do you see a pattern? Will this pattern continue?
Math Talk
Digit Detectives
After writing numbers from 1 to 100, Dinesh wondered how many times he would have written the digit ‘7’!
Among the numbers 1–100, how many times will the digit ‘7’ occur? Among the numbers 1–1000, how many times will the digit ‘7’ occur?
3.5 Pretty Palindromic Patterns
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, 222 are some examples of palindromes using the digits ‘1’, ‘2’, 3’.
Write all possible 3-digit palindromes using these digits.
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 or else 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. Are there numbers for which you do not reach a palindrome at all?
Explore
Will reversing and adding numbers repeatedly, starting with a 2-digit number, always give a palindrome? Explore and find out.*
Math Talk
Puzzle time
tth
hthut
Write the number in words:
I am a 5-digit palindrome. I am an odd number. My ‘t’ digit is double of my ‘u’ digit. My ‘h’ digit is double of my ‘t’ digit. Who am I? _________________
3.6 The Magic Number of Kaprekar
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.
*The answer is yes! For 3-digit numbers the answer is unknown. It is suspected that starting with 196 never yields a palindrome!
Follow these steps and experience the magic for yourselves! Pick any 4-digit number having at least two different digits, say 6382.
Take a 4-digit number.
Make the largest number from these
digits. Call it A.
Make the smallest number from these
Use digits
digits. Call it B.
of C
Subtract B from A. Call it C.
C = A – B
What happens if we continue doing this?
A = 8632 A = 6642 A = 7641 A = B = 2368 B = 2466 B = 1467 B =C = 8632–2368 C = 6642–2466 C = 7641–1467 C = = 6264 = 4176 = 6174
Explore
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?
3.7 Clock and Calendar Numbers
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 of 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, has 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 will also be 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?
TryThis
Figure it Out
1. Pratibha uses the digits ‘4’, ‘7’, ‘3’ and ‘2’, and makes the smallest and largest 4-digit numbers with them: 2347 and 7432. The difference between these two numbers is 7432 – 2347 = 5085. The sum of these two numbers is 9779. Choose 4 - digits to make:
a. the difference between the largest and smallest numbers greater than 5085.
b. the difference between the largest and smallest numbers less than 5085.
c. the sum of the largest and smallest numbers greater than 9779.
d. the sum of the largest and smallest numbers less than 9779.
2. What is the sum of the smallest and largest 5-digit palindrome? What is their difference?
3. The time now is 10:01. How many minutes until the clock shows the next palindromic time? What about the one after that?
4. How many rounds does the number 5683 take to reach the Kaprekar constant?
3.8 Mental Math
Observe the figure below. What can you say about the numbers and the lines drawn?
25,000
38,8003,400
28,00063,000
13,000
61,60019,500
1,500
31,00020,900
60,000
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
Can we make 1,000 using the numbers in the middle? Why not? What about 14,000, 15,000 and 16,000? Yes, it is possible. Explore how. What thousands cannot be made?
Math Talk
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.
39,800 = 40,000 – 800 + 300 + 300
40,0007,000
45,000 =
3001,500
5,900 =
17,500 =
12,000800
21,400 =
Digits and Operations
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.
Figure it Out
1. Write an example for each of the below scenarios whenever
possible.
5-digit + 5-digit to give a 5-digit sum more than 90,250
5-digit + 3-digit to give a 6-digit sum
4-digit + 4-digit to give a 6-digit sum
5-digit + 5-digit to give a 6-digit sum
5-digit + 5-digit to give 18,500
5-digit – 5-digit to give a difference less than 56,503
5-digit – 3-digit to give a 4-digit difference
5-digit − 4-digit to give a 4-digit difference
5-digit − 5-digit to give a 3-digit difference
5-digit − 5-digit to give 91,500
Could you find examples for all the cases? If not, think and discuss what could be the reason. Make other such questions and challenge your classmates.
Math Talk
2. Always, Sometimes, Never?
Below are some statements. Think, explore and find out if each of the statement is ‘Always true’, ‘Only sometimes true’ or ‘Never true’. Why do you think so? Write your reasoning and discuss this with the class.
a. 5-digit number + 5-digit number gives a 5-digit number
b. 4-digit number + 2-digit number gives a 4-digit number
c. 4-digit number + 2-digit number gives a 6-digit number
d. 5-digit number – 5-digit number gives a 5-digit number
e. 5-digit number – 2-digit number gives a 3-digit number
3.9 Playing with Number Patterns
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 one of you used to solve these questions.
a.b.
c.d.
32 32 32 32 32 32 32 32
32 32 32 32 32 32 32 32
32 32 32 32 32 32 32 32
32 32 32 32 32 32 32 32
64 64 6464
64 64 6464
64 64 6464
64 64 6464
15152525
e.f.
2525
25 35
1515
1515
2525
3535
1515
2525
5001000
3535
3535
1515
2525
3535
1515
2525
2525
1515
2525
3535
1515
3.10 An Unsolved Mystery — the Collatz Conjecture!
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?
3.11 Simple Estimation
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.
Figure it Out
We shall do some simple estimates. It is a fun exercise, and you may find it amusing to know the various numbers around us. Remember,
we are not interested in the exact numbers for the following questions. Share your methods of estimation with the class.
1. Steps you would take to walk:a. From the place you are sitting to the classroom doorb. Across the school ground from start to end c. From your classroom door to the school gated. From your school to your home
2. Number of times you blink your eyes or number of breaths you take:a. In a minute b. In an hourc. In a day
3. Name some objects around you that are:a. a few thousand in numberb. more than ten thousand in number
Estimate the answer
Try to guess within 30 seconds. Check your guess with your friends.
1. Number of words in your maths textbook:a. More than 5000
b. Less than 5000
2. Number of students in your school who travel to school by bus: a. More than 200
b. Less than 200
3. Roshan wants to buy milk and 3 types of fruit to make fruit custard for 5 people. He estimates the cost to be ₹100. Do you agree with him? Why or why not?
4. Estimate the distance between Gandhinagar (in Gujarat) to Kohima (in Nagaland). Hint: Look at the map of India to locate these cities.
5. Sheetal is in Grade 6 and says she has spent around 13,000 hours in school till date. Do you agree with her? Why or why not?
6. Earlier, people used to walk long distances as they had no other means of transport. Suppose you walk at your normal pace. Approximately, how long would it take you to go from:
a. Your current location to one of your favourite places nearby. b. Your current location to any neighbouring state’s capital city. c. The southernmost point in India to the northernmost point in India.
7. Make some estimation questions and challenge your classmates!
3.12 Games and Winning Strategies
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 classmate. 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 classmate. 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 own 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!
Figure it Out
1. There is only one supercell (number greater than all its neighbours) in this grid. If you exchange two digits of one of the numbers, there will be 4 supercells. Figure out which digits to swap.
16,20039,34429,765
TryThis
45,30623,609
62,871
19,38150,31938,408
2. How many rounds does your year of birth take to reach the Kaprekar constant?
3. We are the group of 5-digit numbers between 35,000 and 75,000 such that all of our digits are odd. Who is the largest number in our group? Who is the smallest number in our group? Who among us is the closest to 50,000?
4. Estimate the number of holidays you get in a year including
weekends, festivals and vacation. Then, try to get an exact number and see how close your estimate is.
5. Estimate the number of liters a mug, a bucket and an overhead tank can hold.
6. Write one 5-digit number and two 3-digit numbers such that their sum is 18,670.
7. Choose a number between 210 and 390. Create a number pattern similar to those shown in Section 3.9 that will sum up to this number.
8. Recall the sequence of Powers of 2 from Chapter 1, Table 1. Why is the Collatz conjecture correct for all the starting numbers in this sequence?
9. Check if the Collatz Conjecture holds for the starting number 100.
10. Starting with 0, players alternate adding numbers between 1 and 3. The first person to reach 22 wins. What is the winning strategy now?
Su m m a r y
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 difÏcult to solve. Indeed, numerous such problems are still unsolved (for example, Collatz’s Conjecture).
CHAPTER 3 — SOLUTIONS
Q. Think about various situations where we use numbers. List five different situations in which numbers are used. See what your classmates have listed, share, and discuss.
Ans. Five different possible situations in which numbers are used -
1. Time 2. Calendar 3. Counting objects/Marks 4. Measurement of height & weight 5. Money There could many more.
Section 3.1
Page No. 55
Q. What do you think these numbers mean?
Ans. Refer page 56.
Page No. 56
Q1. Can the children rearrange themselves so that the children standing at the ends say ‘2’? Ans. No; There will be no one standing on the other side of the child standing at the end.
Q2. Can we arrange the children in a line so that all would say only 0s? Ans. Yes; All the children in the line should be of same height.
Q3. Can two children standing next to each other say the same number? Ans. Yes; Refer picture on page 55.
Q4. There are 5 children in a group, all of different heights. Can they stand such that four of them say ‘1’ and the last one says ‘0’? Why or why not?
Ans. Yes, they can, if they are standing in ascending order of height.
[1]
Q5. For this group of 5 children, is the sequence 1, 1, 1, 1, 1 possible?
Ans. No; the tallest child at the end cannot say1.
Q6. Is the sequence 0, 1, 2, 1, 0 possible? Why or why not?
Ans. Yes, it is possible.
Q7. How would you rearrange the five children so that the maximum number of children say ‘2’?
Ans. At the most only 2 children can say 2 as given is the following arrangement.
Section 3.2
Page No. 57
Figure it out
Q1. Colour or mark the supercells in the table below.
6828 670 9435 3780 3708 7308 8000 5583 52
Ans.
6828 670 9435 3780 3708 7308 8000 5583 52
[2]
Q2. Fill the table below with only 4-digit numbers such that the supercells are exactly the coloured cells.
5346 1258 9635
Ans. One of the ways could be-5346; 9636.Try more
5346 5347 1000 1258 1100 1200 1300 9635 9636
Q3. Fill the table below such that we get as many supercells as possible. Use numbers between 100 and 1000 without repetitions.
Ans.
110 100 150 130 280 200 230 210 270
Q4. Out of the 9 numbers, how many supercells are there in the table above? ___________
Ans. 5
Q5 Find out how many supercells are possible for different numbers of cells.
Do you notice any pattern? What is the method to fill a given table to get the maximum number of supercells? Explore and share your strategy.
Ans. For even number of cells say,2,4,6,… the number of supercells would be respectively, 2/2 =1,4/2 =2,6/2=3,…
For odd number of cells , say 1,3,5,7,… the number of supercells would be respectively (1+1)/2= 1, (3+1)/2 = 2, (5+1)/2= 3,(7+1)/2 = 4,… To get the maximum number ofsupercells, we have to start by filling the first cell as super cell & then fill alternately.
Q6. Can you fill a supercell table without repeating numbers such that there are no supercells? Why or why not?
Ans. No; the cell which is filled by the greatest number among the given numbers chosen, will become super cell irrespective of its position in the table.
Q7. Will the cell having the largest number in a table always be a supercell? Can the cell having the smallest number in a table be a supercell? Why or why not?
Ans. Yes, the largest number in a table will always be a supercell.
No, the smallest number in a table can never be a supercell as the number in all the adjacent cells will be greater than it.
Q8. Fill a table such that the cell having the second largest number is not a supercell.
Ans. One of the ways could be-
1 2 3 4 5 6 7 9 8
[3]
Q9. Fill a table such that the cell having the second largest number is not a supercell but the second smallest number is a supercell. Is it possible?
Ans. One of the ways is-
Second largest number 8 is not a supercell.
Second smallest number a super cell
2 1 3 4 5 6 7 9 8
Q10. Make other variations of this puzzle and challenge your classmates.
Ans. Some of these could be-
Can you fill the table with 9 cells such that there are more than 5 super cells? Can you fill the table with 9 cells such that there are exactly 4 super cells?
Page No. 58
Q. 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.
Ans. Table 2 (One of the ways) –
96,310 96,301 36,109 39,160 96,103 13,609 60,319 19,306 13,906 10,396 60,193 60,931 10,369 10,963 10,936 69,031
Q. The biggest number in the table is ____________. Ans. The biggest number in the table is 96,310
Q. The smallest even number in the table is ____________. Ans. The smallest even number in the table is 10,396
Q. The smallest number greater than 50,000 in the table is ____________. Ans. The smallest number greater than 50,000 in the table is 60,193.
Section 3.3
Page no.59
Q. 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.
Ans.
1050
2180
8400
9950
3600
5030
2754
9000
1000
2000
3000
4000
5000
6000
8000
10000
7000
[4]
1500
5300
9590
Q. Identify the numbers marked on the number lines below, and label the remaining positions.
(a).
2025
2030
1995
2000
2005
2010
2015
2020
2035
1990
(b).
10000
10001
9995
9996
9997
9998
9999
10002
9993
9994
(c).
15085
15078
15079
15080
15081
15082
15084
15086
15077
15083
(d).
84705
85705
86705
87705
88705
90705
91705
92705
83705
89705
Put a circle around the smallest number and a box around the largest number in each of the sequences above.
Section 3.4
Page no. 60
Q. Find out how many numbers have two digits, three digits, four digits, and five digits:
5 Digit numbers 9
1 Digit numbers
2 Digit numbers
3 Digit numbers
4 Digit numbers
Ans.
1 Digit numbers 2 Digit numbers 3 Digit numbers 4 Digit numbers 5 Digit numbers 9 90 900 9,000 90,000
[5]
Figure it out
Q.1. Digit sum 14
a. Write other numbers whose digits add up to 14.
b. What is the smallest number whose digit sum is 14?
c. What is the largest 5-digit number whose digit sum is 14?
d. How big a number can you form having the digit sum 14? Can you make an even bigger number?
Ans. a. Some such numbers are:248, 653, 356, 815, 833, 12335, 23351. b. 59 c. 95000 d. 95, 9005, 900005, 90000005, 9000000005, 90000000000005 …
Q.3. Calculate the digit sums of 3-digit numbers whose digits are consecutive (for example, 345). Do you see a pattern? Will this pattern continue?
Ans. 123 → 1+2+3 = 6 234 → 2+3+4 = 9 345 → 3+4+5 = 12 456 → 4+5+6 = 15 567 → 5+6+7 = 18 678 → 6+7+8 = 21 789 → 7+8+9 = 24
• Yes, there is a pattern, all the sums are multiples of 3. • No.
Page no. 61
Q. Among the numbers 1–100, how many times will the digit ‘7’ occur? Among the numbers 1–1000, how many times will the digit ‘7’ occur?
Ans.
• 20 times. • 300 times.
[6]
Section 3.5
Page no. 61
Q. Write all possible 3-digit palindromes using these digits.
Ans. Palindromes: 111, 121, 131 222, 212, 232 313, 323, 333 Explore Page no. 62
Q. Will reversing and adding numbers repeatedly, starting with a 2-digit number, always give a palindrome? Explore and find out.*
Some of these are-
47 +74 121 Try more Yes, it will always give a palindrome.
12 +21 33
Puzzle time
Q. I am a 5-digit palindrome.
I am an odd number. My ‘t’ digit is double of my ‘u’ digit. My ‘h’ digit is double of my ‘t’ digit. Who am I? _________________
Ans. tth th h t u
1 2 4 2 1
Twelve thousand four hundred twenty one. Section 3.6 Page no. 63
Q. Carry out these same steps with a few 3-digit numbers. What number will start repeating? Ans.
Take a 3- Digit number say, 321. 321 -123 198
954 -459 495 The number 495 starts repeating. Try for other 3-digit numbers.
981 -189 792
972 -279 693
963 -369 594
954 -459 495
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Section 3.7 Page no. 64 Q. Try and find out all possible times on a 12-hour clock of each of these types. Ans. 4:44 2:22 3:33 10:10 11:11 12:12 09:09 12:21 05:50 10:01 Think of some more! Q. Find some other dates of this form from the past. Ans. 20/04/2004, 20/06/2006, Try for yourself! Q. Find all possible dates of this form from the past. Ans. 01/02/2001, 02/02/2002, Think of some more! Q. 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? Ans. Yes, The calendar repeats itself after 6 years if only one leap year is included in these 6 years. If 2 leap years are included, then it will repeat after 5 years. Page no. 64
Figure it out Q.1. Pratibha uses the digits ‘4’, ‘7’, ‘3’ and ‘2’, and makes the smallest and largest 4-digit numbers with them: 2347 and 7432. The difference between these two numbers is 7432 – 2347 = 5085. The sum of these two numbers is 9779. Choose 4–digits to make:
a. the difference between the largest and smallest numbers greater than 5085. b. the difference between the largest and smallest numbers less than 5085. c. the sum of the largest and smallest numbers greater than 9779. d. the sum of the largest and smallest numbers less than 9779. Ans. Some of the possibilities are–
a. 7431 – 1347 = 6084 b. 7433 – 3347 = 4086 c. 7433 + 3347 = 10780 d. 7431 + 1347 = 8778 Q.2. What is the sum of the smallest and largest 5-digit palindrome? What is their difference? Ans. Smallest 5 digit palindrome = 10001
largest 5 digit palindrome = 99999 Sum = 10001 + 99999 = 110,000 Difference = 99999 – 10001 = 89,998
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Q.3. The time now is 10:01. How many minutes until the clock shows the next palindromic time? What about the one after that? Ans. Time Now → 10:01
Next palindrome time → 11:11 After 1 hr. 10 min = 70 min the clock will show next palindrome time. Next palindrome time = 12:21 which will occur after 2 hr. 20 min = 140 min from 10:01. Q.4. How many rounds does the number 5683 take to reach the Kaprekar constant? Ans. 5683
8653 -3568 5085
8550 -5058 3492
9432 -2349 7083
8730 -3078 5652
6552 -2556 3996
9963 -3699 6264
6642 -2466 4176
7641 -1467 6174
1 2 3 4 5 6 7 8 It will take 8 rounds to reach the Kaprekar constant. Page no. 66 Section 3.8 Q. Can we make 1,000 using the numbers in the middle? Why not? What about 14,000, 15,000 and 16,000? Yes, it is possible. Explore how. What thousands cannot be made? Ans. No; the only number which is smaller than 1000 is 400 and 1000 is not a multiple of 400.
14000 = 1500 × 8 + 400 × 5 = 12000 + 2000 = 14000 15000 = 13000 + 400 × 5 = 13000 + 2000 = 15000 16000 = 1500 × 8 + 400 × 10 = 12000 + 4000 = 16000 Only one thousand (1000) cannot be made. Figure it out Q.1. Write an example for each of the below scenarios whenever possible.
Could you find examples for all the cases? If not, think and discuss what could be the reason. Make other such questions and challenge your classmates. Ans.
• 5 digit + 5 digit > 90,250
45,000 + 45,400 = 90,400 > 90,250 • 5 digit + 3 digit = 6 digit sum
99,999 + 999 = 100,998 • 4 digit + 4 digit = 6 digit sum
Not possible as even the sum of the greatest 4 digit numbers will not give a six digit sum. (9999 + 9999 = 19,998)
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• 5 digit + 5 digit = 6 digit sum
60,000 + 40,000 = 1,00,000 • 5 digit + 5 digit = 18,500
Not possible as smallest 5-digit number is 10,000. If both the numbers are 10,000 then the sum is 20,000, which is more than 18,500. • 5 digit – 5 digit < 56,503
80,000 – 50,000 < 56,503 < 56,503 • 5 digit – 3 digit = 4 digit difference
10,000 – 999 = 9001 • 5 digit – 4 digit = 4 digit difference
12,000 – 2,500 = 9,500 • 5 digit – 5 digit = 3 digit difference
50,999 – 50,000 = 999 • 5 digit – 5 digit = 91,500
Not possible as the difference of the greatest and the smallest 5 digit numbers, the maximum difference, can be 99,999 – 10,000 = 89,999 Some examples of other such questions are-
1. 5 digit + 5 digit = 7 digit sum 2. 4 digit + 4 digit = 2900 More such examples can be made. Q.2. Always, Sometimes, Never?
Below are some statements. Think, explore and find out if each of the statement is ‘Always true’, ‘Only sometimes true’ or ‘Never true’. Why do you think so? Write your reasoning; discuss this with the class. a. 5-digit number + 5-digit number gives a 5-digit number b. 4-digit number + 2-digit number gives a 4-digit number c. 4-digit number + 2-digit number gives a 6-digit number d. 5-digit number – 5-digit number gives a 5-digit number e. 5-digit number – 2-digit number gives a 3-digit number
Ans. a. Only sometimes true.
20,000 + 80,000 = 1,00,000 not a 5 digit number b. Only sometimes true
9,999 + 99 = 10,098 not a 4 digit number c. Never true
9,999 + 99 = 10,098
On adding the greatest 4 digit and the greatest 2 digit numbers, we can reach only 5 digit number 10,098. So there is no possibility of getting a 6 digit number. d. Only sometimes true
Ex. 12,000 – 10,000 = 2,000 5 digit – 5 digit = 4 digit e. Never true
Ex. 10,000 – 99 = 9901
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Even if the greatest 2 digit numbers is subtracted from the smallest 5 digit number, 4 digit number will be obtained. Page no. 69 Section 3.10
Q. 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? Ans. a) 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1
b) 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1
Yes we always reach 1. The even numbers are halved and when we have an odd number we convert it into an even number by multiplying by 3 and adding 1 so that eventually it can be halved again. The smallest even number is 2 so we will reach 1 for sure. Q.3. Name some objects around you that are:
a. a few thousand in number b. more than ten thousand in number Ans. A few thousands: car numbers, 4 digit pin More than ten thousands: Salary, Mobile numbers. Q.3. Roshan wants to buy milk and 3 types of fruit to make fruit custard for 5 people. He estimates the cost to be ₹ 100. Do you agree with him? Why or why not? Ans. Yes, it is possible with less quantity of serving and with purchase of 1-1-1 fruit of each type. However, it is not possible with costly fruits and more quantity of serving. Q.4. Estimate the distance between Gandhinagar (in Gujarat) to Kohima (in Nagaland). Ans. 2500 kilometer Q. 5. Sheetal is in Grade 6 and says she has spent around 13,000 hours in school till date. Do you agree with her? Why or why not? Ans. No, I do not agree with her. There are 6 school hours in a day and around 200 working days in a year.
𝟏𝟑,𝟎𝟎𝟎
𝟔×𝟐𝟎𝟎 = 10.8 years
(Nursery, KG, 1,2,3,4,5,6) 8 years She is in school for 8 years, 13,000 hours is way too high. Q.7. Make some estimation questions and challenge your classmates! Ans. Some such are
• How many students are there in your school? • How many hours does a person sleep in his lifetime on an average?
(More such questions can be made)
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Section 3.12 Page No. 72 Figure it Out Q.1. There is only one supercell (number greater than all its neighbours) in this grid. If you exchange two digits of one of the numbers, there will be 4 supercells. Figure out which digits to swap.
16,200 39,344 29,765
23,609 62,871 45,306
19,381 50,319 38,408
Ans. If I exchange the digits 1 and 6 in the number 62,871 then there will be 4 Super cells.
16,200 39,344 29,765
23,609 12,876 45,306
19,381 50,319 38,408
Q.2. How many rounds does your year of birth take to reach the Kaprekar constant? Ans. Suppose the birth year is 1980, then, -
7641 -1467 6174 It takes 6 rounds. (Try now for your year of birth.) Q.3. We are the group of 5-digit numbers between 35,000 and 75,000 such that all of our digits are odd. Who is the largest number in our group? Who is the smallest number in our group? Who among us is the closest to 50,000? Ans.
9810 -1089 8721
8721 -1278 7443
7443 -3447 3996
9963 -3699 6264
6642 -2466 4176
With repeating digit With non repeating digit Largest number → 73,999
73,951
Smallest number → 35,111 35,179 With repeating digit With non repeating digit Closest to 50000 51,111 51,379 Q.6. Write one 5-digit number and two 3-digit numbers such that their sum is 18,670. Ans. 18000 + 300 + 370 = 18670. Try for more.
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Q.7. Choose a number between 210 and 390. Create a number pattern similar to those shown in Section 3.9 that will sum up to this number. Ans. Number Chosen: 250 25 25 50 50 50 25 25
Or
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Q.8. Recall the sequence of Powers of 2 from Chapter 1, Table 1. Why is the Collatz conjecture correct for all the starting numbers in this sequence? Ans. when we divide 28 = 2x2x2x2x2x2x2x2 by 2 it become 27 and every time you divide by 2 the same will continue happening, until you are left with 2 which when divided by 2 will leave 1. Q.9. Check if the Collatz Conjecture holds for the starting number 100. Ans. 100, 50, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. Q.10 Starting with 0, players alternate adding numbers between 1 and 3. The first person to reach 22 wins. What is the winning strategy now? Ans. Winning strategy is to be the first player.
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