# Real Life Problems Involving H CF and LCM

#### Students’ Learning Outcomes

·         Solve real life problems involving H CF and LCM.

#### Information for Teacher

·         In continuous division process, from two given numbers, any one is taken as divisor and other is as dividend, and after division, now remainder is taken as divisor and the first divisor is taken as dividend and we repeat division process. The process keep continue until we get the remainder ‘0’ hence the last non zero divisor is called G. C. D. or H CF.
·         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 (LCM)
·         H CF and LCM are of great importance in our daily life.
·         We use H CF, when we have to make smaller parts of anything through division.
·         We use LCM, if there are repeated events.
·         Similarly we need the least common multiple to find a common denominator, for adding or subtracting fractions, and to write the fractions into simplest form, we need greatest common divisor.
·         Practice more and more to the students, of finding H CF and LCM mentally.
·         Review some tables to students, especially 12 x 12, 13 x 13, 22 x 22 etc. Because if students are good at tables, they can make correct estimates.

It’s god if these types of special multiplications are reinforced every now and then.

#### Material / Resources

Board, marker/chalk, duster, paper chits on which questions are written, textbook

#### Worm up activity

o   What do we mean by H CF?
o   What are the methods of finding H CF?
o   Which method you like and why?
o   Does anyone know any other new method of finding H CF?
o   Tell me with estimation, what would be the H CF of 60 and 84?
o   What would be the H CF of 30, 24, 16, 42, give estimated answer?
o   What do we mean by LCM?
o   What are the methods of finding LCM?
o   Which method you like and why?
o   Does anyone know any other new method of finding LCM?
o   Tell the students that, today we are going to look at the application or use of H CF and LCM in real life.
o   For example we need the least common multiple to find a common denominator, for adding or subtracting fractions, and to write the fractions into simplest form, we need greatest common divisor.

#### Activity 1

·         With the help of following examples, we can see the use of H CF and LCM.
·         How long a stick can exactly measure 10, 12, 14 meters lengths?
·         As it is clear from the question that the length of the stick should divide exactly these given three lengths.
·         So we have to find H CF, as it is division question.
10 = 2 x 5
12 = 2 x 2 x 3
14 = 2 x 7
·         The H CF of given numbers is 2, so two meter long stick can measure all the lengths exactly.
·         Ali takes milk after every 12 hour and eats bread after every 8 hour. How often does he need to take both foods at once?
·         To answer this, you need to find the LCM of 12 and 8 which is 24.
·         It means after every 24 hours Ali takes both foods at the same time.
·         With these examples, we have seen that H CF and LCM has application in our daily life as well.

#### Activity 2

·         Method to solve word problems of H CF and LCM.
o   Keep following in mind.
o   Use the KW L method for approaching all problems.
o   K mean: What do you know?
o   W mean: What do you want to know?
o   L mean: What did you learn?
o   Repeat KW L two or three times with the help of the students.
·         If it is a H CF problem:
o   What is the question asking us?
o   Do we have to split things into smaller sections?
o   Are we trying to figure out how many people we can invite?
o   Are we trying to arrange something into groups or rows?

#### Activity 3

·         Write following problem on board and ask students to write on their copies.
·         Shillalah has two pieces of cloth. One piece is 72 inches wide and the other piece is 90 inches wide. She wants to cut both pieces into strips of equal width that are as wide as possible. How wide should she cut the strips?
·         Ask students to underline the important points as per KW L.
·         Allocate time 5 minutes.
·         Let them think and do independently.
·         Allow them to share in pairs.
·         After collecting their answers, explain on the board as follows.
·         K: the pieces of cloth are 72 and 90 inches wide.
·         W: how wide should she cut the strips so that they are the largest possible equal lengths?
·         L: this problem can be solved using H CF because we are cutting or ‘dividing’ the strips of cloth into smaller pieces (factors) of 72 and 90.
·         Hence we have seen that where there is dividing or making its smaller, we use H CF.
·         Find the H CF of 72 and 90. When you are done do thumb up.
H CF = 2 x 3 x 3 = 18
·         So Shillalah should cut each piece to be 18 inches wide.

#### Activity 4

·         If it is a LCM problem (being the discussion again).
·         What is the question asking us?
·         Do we have an event that is or will be repeating over and over?
·         Are we trying to figure out when something will happen again at the same time?
·         Write the following problem on the board and ask students to write as well.
·         Bilabial exercises every 12thday and Ismail every 8th day. Bilabial and Ismail both exercised today. How many days will it be until they exercise together again?
·         Ask students to underline the important points as per KW L.
·         Allocate time 5 minutes.
·         Let them think and do independently.
·         Allow them to share in pairs.
·         After collecting their answers, explain on the board as follows.
·         K: Bilabial exercises every 12thday and Ismail every 8th day and they both exercised today.
·         W: How many days it until they will both exercise on the same day again.
·         L: This problem can be solved using LCM. Because we are trying to figure out when will be the next time they are exercising together. Further we have seen that where there are repeated events, we use LCM.
·         Ask to find the LCM of 12 and 8.
LCM = 2 x 2 x 2 x 3 = 24
·         Hence Bilabial and Ismail would exercise after every 24 days.

#### Activity 5

·         Divide the students into groups of mix ability.
·         Now provide each group one paper chit/slip of H CF question and other chit/slip of LCM question.
·         Give students 10 minutes to solve.
·         Now one by one call any two students from each group and ask them to solve and explain the given questions on board.
·         At the end appreciate students through clapping.

#### Sum up / Conclusion

·         H CF and LCM are of great importance in our daily life.
·         We use H CF, when we have to make smaller parts of anything through division.
·         We use LCM, if there are repeated events.
·         Similarly we need the least common multiple to find a common denominator, for adding or subtracting fractions and to write the fractions into simplest form, we need greatest common divisor.
·         Use the KW L method for approaching all problems.
·         K mean: What do you know?
·         W mean: What do you want to know?
·         L Mean: What did you learn?

#### Assessment

·         Write following questions on board.
o   Mrs. Hess-en has 120 crayons and 30 pieces of paper to give to her students. What is the largest number of students she can have in her class so that each student gets equal number of crayons and equal number of paper (H CF)
o   Amara is making on a game board that is 16 inches by 24 inches. She wants to use square tiles. What is the largest tile she can use? (H CF)
o   Two bikers are riding a circular path. The first rider completes a round in 12 minutes. The second rider completes a round in 18 minutes. If they both started at the same place and time and go in the same direction, after how many time will they meet again at the starting point? (LCM)s
o   Ali has 8 inches pieces of toy train track and Ahmad has 18 inches pieces of train track. How many of each piece would each child need to build tracks that are equal in length? (H CF)
o   Now read questions one by one and ask students to think and tell this question will solve with the help of H CF or LCM?
o   After taking response from students, tell them that which is the H CF question and which is the LCM.
o   Now ask students to solve these questions independently.
o   Collect copies from students and check.