Greatest Common Factor and Least Common Multiple

GREATEST COMMON FACTOR AND LEAST COMMON MULTIPLE

INTRODUCTION

The objective for this lesson on Greatest Common Factor and Least Common Multiple is, the student will investigate common factors and least common multiples.

The skills students should have in order to help them in this lesson include, Multiplication, Division, Factors and Multiples.

We will have three essential questions that will be guiding our lesson. Number One, what is the difference between greatest common factor (GCF and least common multiple (LCM)? Number Two, what is the most important word in greatest common factor? Explain. Number Three, what is the most important word in least common multiple? Explain.

Begin by completing the warm-up on multiplication, and review of factors, products and multiples to prepare for the lesson on Greatest Common Factor and Least Common Multiple.

SOLVE PROBLEM – INTRODUCTION

The SOLVE problem for this lesson is, Maddie and Elli are selling tickets to the school play. They cost three dollars and fifty cents each. They sold tickets for several hours over the weekend. If Maddie sold five tickets per hour and Elli sold eight tickets every other hour, how many tickets would they need to sell to have sold exactly the same amount of tickets?

In Step S, we Study the Problem. First we need to identify where the question is located within the problem and underline the question. The question for this problem is, how many tickets would they need to sell to have sold exactly the same amount of tickets?

Now that we’ve identified the question, we need to put this question in our own words in the form of a statement. This problem is asking me to find the number of tickets Maddie and Elli have to sell to have sold the same amount of tickets.

During this lesson we will be working with Greatest Common Factor and Least Common Multiple to complete this SOLVE problem at the end of the lesson.

MODELING WITH GREATEST COMMON FACTOR - CONCRETE

Identify the multiplication facts that will have a product of four. One times four is equal to four and two times two is equal to four.

Explain the meaning of one times four equal four. There is one group of four items which is equal to four.

What division facts are related to one times four equals four? Four divided by one equals four or four divided by four equals one.

What are the factors of one times four equal four? One and four

Explain how you know those are the factors. Both one and four divide evenly into four.

Work together to write a definition of factors. A factor is a number that divides evenly into another number. A factor is a number that can be multiplied by another number to equal a product.

All students with a blue shirt, please stand.

What do these students have that are the same? Blue Shirts

Thank you, please have a seat.

All students who are wearing sneakers, please stand.

What do these students have that are the same? Sneakers

Thank you, please have a seat.

When all the students stood who had a blue shirt, they had something in common. What was that characteristic? A blue shirt.

What is the GCF or Greatest Common Factor of twelve and eighteen? Let’s apply the same idea to numbers.

Look at the value of twelve. Place a red centimeter cube on each of the factors of twelve. One, two, three, four, six and twelve.

What are the factors that you have covered? One, two, three, four, six and twelve.

Now, place centimeter cubes on all the factors of eighteen. One, two, three, six, nine and eighteen.

What do you notice about the first factor? It is the same as the first factor of twelve.

Look at the factors and determine which ones are common to both twelve and eighteen. Which factors have both color cubes? They are one, two, three, and six.

What is the GCF or Greatest Common Factor of twelve and eighteen? Identify the greatest value with both colors. Six

What is the greatest value or factor that twelve and eighteen have in common or share? Six

This means that six is the greatest common factor or GCF of twelve and eighteen.

What is a factor? A number that divides evenly into another number, a number that can be multiplied by another number to equal a product.

Explain the meaning of the word common. Something that is shared.

Remember that the students who had blue shirts and the students wearing sneakers had something in common.

What does the phrase greatest common factor mean? The greatest common factor shared by two numbers.

GREATEST COMMON FACTOR – PICTORIAL

Let’s begin by drawing a vertical red line through each of the factors of twelve. One, two, three, four, six and twelve.

Identify the values that we have marked in red on the chart. One, two, three, four, six and twelve. Write the factors of twelve in the graphic organizer.

Now, let’s use a slanted blue line to mark off the factors of thirty. One, two, three, five, six, ten, fifteen and thirty.

Identify the values that we have marked in blue on the chart. One, two, three, five, six, ten, fifteen and thirty. Write the factors of thirty in the graphic organizer.

How did we identify the common factors when we were using the centimeter cubes? The common factors of twelve and eighteen had both color cubes.

How can we identify the common factors in the hundreds chart? The values will have both colors in the box.

If we look at the graphic organizer, how can we recognize the common factors? They are the numbers that are in both rows.

Circle all of the common factors in the graphic organizer. What values did you circle? One, two, three and six

What relationship do you see between the centimeter cube model, the model with the red and blue marks, and the graphic organizer? They all have the same values marked.

Identify the common factors of twelve and thirty. One, two, three and six

What is the greatest common factor? Six

GREATEST COMMON FACTOR – ABSTRACT

Find the greatest common factor of twenty four and thirty two.

We will not be using a hundred board, but will use the graphic organizer we used in the previous problems to list the factors of twenty four and thirty two.

Identify the factors of twenty four. One, two, three, four, six, eight, twelve and twenty four

Identify the factors of thirty two. One, two, four, eight, sixteen and thirty two

Circle and then list the common factors. One, two, four and eight

What is the greatest common factor of twenty four and thirty two? Eight

MODELING WITH LEAST COMMON MULTIPLE – CONCRETE

What is the LCM or least common multiple of four and five?

What is the meaning of the word “multiple?” Having to do with multiplication

Place a red centimeter cube on the number four in the chart.

If we are counting by fours, what is the next value we would cover? Eight

Explain how you know this? Four plus four equals eight.

If we are counting by fours, what is the next value we would cover? Twelve

Explain your answer. Eight plus four equals twelve

Let’s look at the three values that we have covered: four, eight and twelve

Identify them as groups of four: Four is one group of four; eight is two groups of four; twelve is three groups of four.

Explain how we can write these as multiplication facts. Four times one equal four; four times two equals eight; four times three equals twelve.

What is the next square we should cover? Use the next multiplication factor of four to defend you answer. Sixteen because four times four equals sixteen.

What is the next square we should cover? Use the next multiplication fact of four to defend your answer. Twenty because four times five equals twenty.

What do the values of four, eight, twelve sixteen and twenty have in common? They are all multiples of four.

What does it mean to be a multiple of four? Each value will divide evenly by four: Twenty divided by four equal five; sixteen divided by four equals four; twelve divided by four equals three, eight divided by four equals two, and four divided by four equals one.

Another way to express that they can be divided evenly by four is to say that all those values are divisible by four.

Let’s cover all the multiplies of four.

What multiples should we make sure we have covered? Four, eight, twelve, sixteen and twenty

How can we identify the multiples of five? Multiply five times one, five times two, and so on.

Place a blue centimeter cube on the multiples of five up to twenty. Five, ten, fifteen, twenty

What multiples of five did you cover? Five, ten, fifteen and twenty

What does this mean? All those value are divisible by five.

Therefore, what is the LCM or least common multiple of four and five? Which of the multiples has both color cubes? Twenty

What does this mean? Twenty is the least common multiple of four and five.

LEAST COMMON MULTIPLE – PICTORIAL

Go back to your previous page and place a red centimeter cube on the multiples of four and a blue centimeter cube on the multiples of five. We have four, eight, twelve, sixteen and twenty and then we have the first five multiples of five. Five, ten, fifteen, twenty and twenty five.

Now, draw a red slanted line on the values that were covered with a red centimeter cube. Four, eight, twelve, sixteen and twenty.

Identify the values that we have marked in red on the chart. Four, eight, twelve, sixteen and twenty. Write them in the graphic organizer.

Now draw different diagonal lines (top left to bottom right) with a different color pencil to show the multiples of five through twenty five.

Identify the values that we have marked in blue on the chart. Five, ten, fifteen, twenty and twenty five. Write them in the graphic organizer.

Now, circle and list the multiple that is common and identify the least common multiple of four and five. Twenty

LEAST COMMON MULTIPLE – ABSTRACT

Find the least common multiple for four and six.

We will not be using a hundreds board, but will use the graphic organizer we used in the previous problem to list the multiplies of four and five.

Identify the multiples of four. Four, eight, twelve, sixteen, twenty, twenty four and twenty eight

Identify the multiples of six. Six, twelve, eighteen, twenty four, thirty, thirty six and forty two

Circle and then list the common multiples. Twelve and twenty four

What is the least common multiple of four and six? Twelve

APPLYING AND DISTRIBUTIVE PROPERTY WITH FACTORS

Example One: Forty eight plus thirty six

What are some strategies for finding the sum? We can apply what we learned about factors and multiples to find the sum of forty eight plus thirty six.

Find the factors of both forty eight and thirty six.

What are the factors of forty eight? One, two, three, four, six, eight, twelve, sixteen, twenty four and forty eight.

What are the factors of thirty six? One, two, three, four, six, nine, twelve, eighteen and thirty six

Find the greatest common factor. Twelve

What is the factor pair for forty eight when twelve is one of the factors? Twelve and four

What is the factor pair for thirty six when twelve is one of the factors? Twelve and three

What is the common factor? Twelve

How can we write the value of forty eight as a multiplication fact with twelve as a factor? Forty eight is equal to twelve times four.

How can we write the value of thirty six as a multiplication fact with twelve as a factor? Thirty six is equal to twelve times three.

What value do both of the facts have in common? Twelve

If we look at the two facts and factors without the common factor of twelve, what is remaining? Four and three.

What is another way we can write four and three using an operation symbol? Four plus three

What is the sum of four plus three? Seven

How can we rewrite twelve times four plus three? Explain your answer. Twelve times seven. We added the two values within the parentheses.

What is the sum of forty eight plus thirty six? Eighty four

What is the product of twelve times seven? Eighty four

What is true about the sum of forty eight plus thirty six and the product of twelve and seven? They are equivalent.

We can use the greatest common factor and the distributive property to find the sum of numbers.

SOLVE PROBLEM – CLOSURE

Now let’s go back to our SOLVE problem from the beginning of the lesson. Maddie and Elli are selling tickets to the school play. They cost three dollars and fifty cents each. They sold tickets for several hours over the weekend. If Maddie sold five tickets per hour and Elli sold eight tickets every other hour, how many tickets would they need to sell to have sold exactly the same amount of tickets?

In the S Step, we Study the Problem. Underline the question and complete this statement. This problem is asking me to find the number of tickets Maddie and Elli have to sell to have sold the same amount of tickets.

O, Organize the Facts. First identify the facts. Maddie and Elli are selling tickets to the school play. / Place a strike mark or a vertical line at the end of each fact. They cost three dollars and fifty cents each. / They sold tickets for several hours over the weekend./ If Maddie sold five tickets per hour/ and Elli sold eight tickets every other hour,/ how many tickets would they need to sell to have sold exactly the same amount of tickets?

After we identify the facts, then we eliminate the unnecessary facts. We go back to our original problem. If the fact will help us determine how many tickets they would need to sell in order to sell the same amount, then we keep that fact. If not we can draw a strike mark through that fact.

After that we list the necessary facts. Maddie sold five tickets per hours and Elli sold eight tickets every other hour.

L, Line Up a Plan. Write in words what your plan of action will be. Set up a graphic organizer to show multiples of five and multiples of eight. Find the least common multiple.

Choose an operation or operations. Multiplication, we’ll be using multiples.

V, Verify Your Plan with Action. First estimate your answer. Our estimate here is about forty. We multiplied eight times five.

Then carry out your plan. We draw a table where we write the multiples of five. Then we write the multiples of eight until we have a common multiple.

E, Examine Your Results.

Does your answer make sense? Compare your answer to the question. Yes, because we are looking for the identical number of tickets both students will sell.

Is your answer reasonable? Compare your answer to the estimate. Yes, because it matches our estimate of forty.

Is your answer accurate? Check your work. Yes.

Write your answer in a complete sentence. Maddie and Elli will have to sell forty tickets to have the same amount of tickets sold.

CLOSURE

Now let’s go back and discuss the essential questions from this lesson.

Our first question was, what is the difference between greatest common factor (GCF) and least common multiple (LCM)? The greatest common factor is the largest number that is a factor of two or more whole numbers. The least common multiple is the smallest number that is a multiple of two or more whole numbers. It cannot be zero.

What is the most important word in greatest common factor? Explain. “Greatest” is the most important word because we are looking for the largest number that is a factor of two or more whole numbers.

What is the most important word in least common multiple? Explain. “Least” is the most important word because we are looking for the smallest number that is a multiple of two or more whole numbers – except for zero.