Students’ Learning Outcomes
· Find LCM of four numbers, up to 2 digits, using prime factorization method.
· Find LCM of four numbers, up to 2 digits, using division method.
Information for Teacher
· Since the students have done both factorization and division method for finding H CF, so they won’t find this difficult.
· First go with three numbers questions and explain both methods, then individual or group work and then discuss on four numbers questions.
· When any number is used to multiply with the set of natural numbers {1, 2, 3, ……}, the product of that number with each of the natural numbers is called multiples of that number, e.g. the multiples of three are : 3(1), 3(2(, 3(3), 3(4(, …. Which are respectively equal to 3, 6, 9, 12……..
· The number/multiple among the common multiples of two or more numbers is called the least common multiple.
· Least common multiple is abbreviated as ‘LCM’.
· Find LCM of two or more numbers through multiples.
· In this method we find few multiples of all given numbers, then write the common multiples separately and the smallest of these common multiples is called LCM.
· Find LCM of 6, 8 and 12 by finding multiples.
· Find LCM of two or more numbers through prime factorization.
· In prime factorization method the product of common and uncommon factors is called LCM.
· In this method, first write prime factorization of each given number, then in each factorization, the factors repeating, write them in exponential form.
· Separate all highest power factors from all factorizations.
· Now the product of all factors is called LCM.
· Find LCM of 72, 24, 96 and 12 by using prime factorization.
· LCM of the given numbers = in all factorizations, product of all such prime factors which having highest exponent.
· Instead of writing factors in exponential form, you can identify all the common factors after factorization of each number, and then product of all common and uncommon factors is called LCM.
· To find LCM while writing the common factors, keep in mind that it is not necessary that common factors are common in all given numbers, if they are common in at least two of given numbers then these are also consider in common numbers.
LCM of given numbers = product of common factors (including factors common in at least two numbers) and uncommon factors.
= 2 x 2 x 2 x 3 x 2 x 2 x 3
LCM = 288
· Find the LCM of two or more numbers using division method.
· To find LCM of two or more numbers using division method, we continue the division process until below all numbers we have 1’s as quotient.
· The divisors may be prime and composite, but try to divide with smallest number.
· In this method the product of divisors of the numbers is called LCM.
· Like finding H CF by using division method (Ladder method) it is not necessary that all the numbers will divide with one divisor. I.e. the number which is not divisible by that divisor, then simply write that number down.
· Find LCM of 72, 24, 96 and 12 using division method.
· During teaching the lesson, teacher should concern with text book, when and where necessary in all steps.
Material / Resources
Board, marker/chalk, duster, text book
Worm up activity
· Divide the students in pairs.
· Inform students that you can discuss with each other about factors of numbers and prime factorization for one minute.
· They will need to select which student will begin first.
· Provide students with the three or four questions to help in their discussion.
· For discussion in pairs, allocate time two minute.
· At one minute, instruct students to switch. At this point, the other partner begins talking.
· It is OK for the second student to repeat some of the things the first student said.
· However, they are encouraged to try and think of new information to share.
Two minute sample questions:
o What are factors?
o Can you tell two things about factors?
o What are prime or composite factors?
o How would you find the factors of a number?
o What is meant by factorization?
o What is prime factorization?
Development
Activity 1
· Inform the students that the numbers divide any number exactly is called factors of this number. E.g. all possible factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30.
· In these factors 2, 3, 5 are prime factors while 6, 10, 15, 30 are composite factors.
· Now tell the students that to write any number as a product of its factors is called factorization. E.g. number 30 can be expressed as a product of its factors in different ways. I.e. following are the factorizations of 30, 30 = 1 x 30 or = 2 x 15 or =3 x 10 or = 5 x 6 or = 2 x 3 x 5.
· Similarly to express any number as a product of its prime factors is called prime factorization. As according to above example, prime factorization of 30 = 2 x 3 x 5.
· All possible factors of 18 are 1, 2, 3, 6, 9, and 18.
· In these factors 2, 3 are prime factors while, 6, 9, 18 are composite factors.
· Number 18 can be expressed as a product of its factors in different ways. I.e. following is the factorization of 18, 18 = 1 x 18 or =2 x 9 or = 3 x 6.
· Similarly prime factorization of 18 = 2 x 3 x 3
· A prime factorization of a number can be written in the exponential form.
E.g. 18 = 2 x 3 x 3 = 2 x 3^{2 }
36 = 2 x 2 x 3 x 3 = 2^{2}x 3^{2}
Activity 2
· Are you ready for something new but as interesting as H CF was.
· OK tell what do we mean by the word product?
· Write 3, 4, and 6 on board and tell the students we are interested in the smallest number that is divisible by 3, 4 and 6.
· If we should find the product of all three numbers i.e. we multiply 3 x 4 x 6 = 72. It seems logical that 72 should be easily divisible by 3, 4 and 6 since 72 is the product of these numbers.
· Ask the students, “Is 72 the smallest number that is divisible by 3, 4 and 6?”
· After collecting response from students tells that the answer is “no”, since 36, which is one-half of 72, is also divisible by 3, 4 and 6.
· Conclude that such a smallest number that is divisible by all given numbers is called LCM.
Activity 3
· Write on the board 42, 96 and 24 or any other question from text book.
· Ask the students to write the given numbers as product of prime factors. E.g. 2 x 2 x 3 etc.
· Now write in exponential form, the numbers which are repeating again and again in each factorization.
· After few minutes, teacher does them on the board as done in the example given in “information for teacher”.
· Now ask them to write the highest power factors as product. E.g. 25 x 32.
· Give few minutes to try, and then ask them to find answer. Thumb up who does it.
· Now do it on the board and explain according to the method given in “information for teacher”.
· Conclude that we find LCM of these numbers using prime factorization.
· Repeat this concept with the help of another example.
· If students feel difficult to write in exponential form and finding LCM, then ask them to write all the common factors and uncommon factors as product and then find LCM.
· Now tell the students, how to solve the question according to this method.
· For practice give students in groups, one or two-digit three numbers than four numbers questions and ask them to find LCM using prime factorization.
Activity 4
· Ask students, do you remember the short cuts to check whether the number is divisible by 2, 3, and 5 or 10?
· After taking their answer, tell the divisibility test rules to all.
· Divisible by 2 if the last digit (ones digit) is 0, 2, 4, 6 or 8 example 12345.
· Divisible by 5 if the sum of digits in the number is divisible by 3. Example 12345.
· Divisible by 5 if the last digit (ones digit) is 0 or 5 example 2345.
· Divisible by 10 if the last digit (ones digit) is ‘0’ example 99990
· Tell them that this will help them while doing LCM with division method.
Activity 5
· Write 3, 4 and 6 on board and solve with the help of students.
· Now start dividing each number by the smallest number like 3, write their answer (quotient) below these numbers.
· If any of the numbers is not divisible by 3, simply write that number down.
· Again start dividing by any smallest number like 2 and then, if any of the numbers is not divisible by it, write that number down.
· Continues this process until in the last line, below each number we have all 1’s as quotient.
· Now by multiplying all the divisors we get LCM.
· Conclude that we have found the LCM of these numbers by division method.
· Repeat this concept with the help of another example.
· For practice give students in groups, one or two-digit three numbers than four numbers questions and ask them to find LCM using division method.
Activity 6
· Assign one question to each student and ask to find LCM by using both methods.
· Allocate time.
· Ask students to check each other work and correct the mistakes.
· Guide the students in question solving.
Sum up / Conclusion
· Least common multiple is abbreviated as ‘LCM’.
· Recap the method of finding LCM using prime factorization.
· Recap the following method of finding LCM using division.
· We write down the numbers horizontally.
· We draw a vertical line to the left of the numbers and draw horizontal line under the numbers.
· We than look for the smallest number that can divide any of the numbers exactly.(try to select the prime divisor)
· If any of the numbers is divisible by divisor, writing down the answer of that division below line. Similarly if any of the numbers is not divisible by divisor, write that number down below the line.
· Then we draw another horizontal line and repeat this process.
· When you get all 1’s as quotient in a horizontal line, then multiply all the divisors that are vertical, and the product called LCM.
· Solve question on board according to both methods.
Assessment
· Find LCM of 48, 24, 16 and 72 by using prime factorization method.
· Find LCM of 30, 18, 68, 8 by using division method.
Follow up
Questions | By using prime factorization | By using division |
· Draw the above said table on board, write questions and ask students to find LCM by both methods.