Patterns in Products
Roxie and Estu are playing with multiplication. They encounter an interesting technique for multiplying a number by 10, 100, 1000, and so on.
A Multiplication Shortcut
Roxie evaluated 116 × 5 as follows:
116 × 5 = 116 × 10/2 (2 cancels out 116 by 58) = 58 × 10 = 580.
Estu evaluated 824 × 25 as follows:
824 × 25 = 824 × 100/4 (4 cancels out 824 by 206) = 20600.
Using the meaning of multiplication and division, can you explain why multiplying by 5 is the same as dividing by 2 and multiplying by 10?
Figure it Out
Find quick ways to calculate these products: (a) 2 × 1768 × 50
(b) 72 × 125 [Hint: 125 = 1000 8 ] (c) 125 × 40 × 8 × 25Calculate these products quickly. (a) 25 × 12 = _____________ (b) 25 × 240 = _____________ (c) 250 × 120 = _____________ (d) 2500 × 12 =_____________ (e) ×= 120000000
How Long is the Product?
In each of the following boxes, the multiplications produce interesting patterns. Evaluate them to find the pattern. Extend the multiplications based on the observed pattern.
11 × 11 = 111 × 111 = 1111 × 1111 =
66 × 61 = 666 × 661 = 6666 × 6661 =
3 × 5 = 33 × 35 = 333 × 335 =
101 × 101 = 102 × 102 = 103 × 103 =
Observe the number of digits in the two numbers being multiplied and their product in each case. Is there any connection between the numbers being multiplied and the number of digits in their product?
Roxie says that the product of two 2-digit numbers can only be a 3- or a 4-digit number. Is she correct?
Should we try all possible multiplications with 2-digit numbers to tell whether Roxie’s claim is true? Or is there a better way to find out?
She explains her reasoning: “We want to know about the number of digits in the product of two 2-digit numbers. To know the smallest such product I took 10 × 10, so all other products will be greater than 100.
To know the greatest such product I multiplied the smallest 3-digit numbers (100 × 100) to get 10,000; so the product of all the 2-digit multiplications will be less than 10,000.”
Can multiplying a 3-digit number with another 3-digit number give a 4-digit number?
Can multiplying a 4-digit number with a 2-digit number give a 5-digit number?
Observe the multiplication statements below. Do you notice any patterns? See if this pattern extends for other numbers as well.
Fascinating Facts about Large Numbers
Some interesting facts about large numbers are hidden below. Calculate the product or quotient to uncover the facts. Once you find the product or quotient, read the number in both Indian and American naming systems. Share your thoughts and questions about the fact with the class after you discover each number.
1250 × 380
is the number of kīrtanas composed by Purandaradāsa according to legends. Purandaradāsa was a composer and singer in the 15th century. His kīrtanas spanned social reform, bhakti and spirituality. He systematised methods for teaching Carnatic music which is followed to the present day.
How many years did he live to compose so many songs? At what age did he start composing songs?
If he composed 4,75,000 songs, how many songs per year did he have to compose?
2100 × 70,000
is the approximate distance
in kilometers, between the Earth
and the Sun. This distance keeps varying throughout the year. The farthest distance is about 152 million kilometers.
How did they measure the distance between the Earth and the Sun?
6400 × 62,500
is the average number of litres of water the Amazon river discharges into the Atlantic Ocean every second. The river's flow into the Atlantic is so much that drinkable freshwater is found even 160 kilometers into the open sea.
As you did before, divide the given numbers to uncover interesting facts about division. Share your thoughts and questions with the class after you uncover each number.
13,95,000 ÷ 150 =
is the distance (in kms) of the longest single-train journey in the world. The train runs in Russia between Moscow and Vladivostok. The duration of this journey is about 7 days. The longest train route in India is from Dibrugarh in Assam to Kanyakumari in Tamil Nadu; it covers 4219 kms in about 76 hours.
Adult blue whales can weigh more than 10,50,00,000 ÷ 700 =
52,00,00,00,000 ÷ 130 =
Large Number Fact In a single gram of healthy soil there can be 100 million to 1 billion bacteria and 1 lakh to 1 million fungi, which can support plants’ growth and health.
Share such large-number facts you know / come across with your class.