Level G Lesson 9
Divide Fractions
The objective for lesson 9 is the student will work with division of fractions. We will have three essential questions that will be guiding our lesson. Question 1, why is it important to know how to build a model for division of fractions? Question 2, what does a division sentence mean? Question 3, what are the steps for dividing a fraction by a fraction?
The SOLVE problem for this lesson is, Ms. Sherrill has three fourths of a yard of red material. She needs to cut the material into pieces that are one eighth of a yard in length. How many pieces of material will she have? We’re going to S the problem and we’re going to start by underlining the question. How many pieces of material will she have? The second part of S is to complete the sentence this problem is asking me to find, the number of pieces of material.
The first problem we are going to look at is 2 divided by one half. The meaning of this problem is, how many groups of one half are in 2 whole units? We’re going to model this problem using our fraction strips a manipulative model and I’m going to start by representing the first fraction. I have two whole units. In a division problem the first number is called a dividend. The second number is called the divisor. The divisor tells us what section we’re going to divide the dividend into. So this time our divisor is one half. So we’re going to divide our whole units into groups of one-half. There’s my first whole unit divided into one half and my second whole unit divided into groups of one half. When I add those 1, 2, 3, 4, I have 4 groups of one half in 2 whole units. I’m going to go up here and now I’m going to model it pictorially. I have 1 group, 2 whole groups and I’m going to divide it into groups of
One half unit. I have 1, 2, 3, 4, 2 divided by one half is equal to 4. The second problem we’re going to model is 2 divided by one third. The meaning of this problem is, how many groups of one third are in 2 whole units? We’re going to model this problem by following our fraction strip by starting with our dividend which is two whole units. We have one unit and then we have a second unit. We’re going to divide our whole units into groups of one third. By one third, I take my green fraction strip, I have 3 units in 1 whole unit and I have 3 one third units in the second unit. So for 2 whole units, I have 1, 2, 3, 4, 5, 6, one third units. I can also model this pictorially by representing my first fraction, my dividend, and dividing it into groups of one third. I have 1 group, 2 groups, 3 groups, 4 groups, 5, groups and a total of 6 groups, 2 divided by one third is 6.
We’re now going to model dividing fractions by another fraction. Our problem is one third divided by one sixth. The meaning of this is, how many groups of one sixth are in one third? In division we represent our first number our dividend with our fraction strip one third unit. We then need to find out how many groups of our one sixth we can fit into that one third unit. We take our fraction strips and we see that there are 2 groups of one sixth in one third. This problem can be modeled pictorially by drawing the one third fraction bar, and the 2 groups of one sixth underneath. I have a total of 2 groups, 2 groups of one sixth.
The next problem we’re going to model is two thirds divided by one twelfth. The meaning of the problem is, how many groups of one twelfth are in two thirds? When we are modeling division multiplication we represent our first fraction. I have my two thirds or 2, one third units. I need to divide that into groups of one twelfth. So I’m going to place my pink fraction strips from my manipulative kit underneath my one third units to see how many groups of one twelfth are in two thirds. I have 1, 2, 3, 4, 5, 6, 7, 8. I can model this pictorially by drawing my 2 one third fraction strips and then drawing the one twelfth units underneath to show that there are 8. I’m then going to add those up for a total of 8. How many groups of one twelfth are in two thirds unit, 8 groups of one twelfth.
We are now gong to model the problem three fourths divided by one half. In division we always start with our first fraction or our dividend. The meaning of or problem is, how many groups of one half are in three fourths? I’m going to take my one half units and place them underneath my one fourth units. You can see that we have a second brown strip that goes beyond our 3, one fourth units. What we’re going to do is we’re going to take the one half unit and underneath that I’m going to place the legal trade of two fourths. Now I have completely covered my three fourths unit, 1 full one half unit covers my two fourths and one half of my brown unit covers my third fourth in this picture. Because we’ve divided the one half into 2 sections, 1 out of the 2 makes up the third fourth. When we go up to the pictorial model we have our 3 fourths and we are dividing into groups of one half. Our one half extends over the last fourth so we divide the one half into 2 pieces, one out of those two pieces or one half of the one half completes our model. So we have 1 whole group of one half, and then one half of a group which is our one fourth. How many groups of one half are in three fourths? 1 and one half groups of one half.
We are going to talk about a method of dividing fractions without using models. We are going to be learning about reciprocals. One fourth and 4 are reciprocals because they have a product of 1. Any whole number can b e written as a fraction by placing it over one, so the 4 whole number can be written as an improper faction of 4 over 1. A number is considered to be a reciprocal if the product of 2 fractions is equal to 1. We found the reciprocal of one fourth by inverting the fraction, that is flipping the fraction so that the denominator is now the numerator and the numerator is now the denominator. In example 2, we’re going to look at one fifth and 5. These 2 values are reciprocals. Remember we are going to first change our whole number to an improper fraction by writing it over 1. Any whole number can be written as a fraction by placing it over 1 as the denominator because 5 divided by 1 is still 5. When we multiply one fifth times 5 over 1 we see that our product is 1. These 2 numbers are reciprocals.
The last example we’re going to look at is example 3. The question is what is the reciprocal of two thirds. The reciprocal of two thirds is 3 halves. We have inverted or flipped our fraction so that the denominator is now the numerator and the numerator is now the denominator. We know that these two values are reciprocals because when we multiply them because our product is 1.
We’re going to model a division of fraction problems using reciprocals. Remember a reciprocal is any two pairs of fractions that are multiplied with a product of 1. We find the reciprocal of the fractions by inverting it. We are going to start our problem
three fourths divided by five sixth by rewriting it vertically. Remember we have two fractions here but that the fraction line is also an indicator of the operation of division, 3 / 4 is actually 3 divided by 4, five sixths is actually 5 divided by 6, and we can change this division sign to the vertical model. When we are dividing two values we need to get a denominator of 1 here. We can do that by multiplying five sixths by it’s reciprocal which is six fifths. Remember you find the reciprocal by inverting the fraction. If we’re going to multiply by six fifths we also to have to multiply our numerator by six fifths, and we can do that because if we look at these two values six fifths divided by six fifths is the value of 1. If we divide something by 1 we do not change the final value. We’re going to go ahead and multiply our numerator’s, 3 times 6 is 18, 4 times 5 is 20, 5 times 6 is 30, 6 times 5 is 30, 30 over 30 is equivalent to 1. Our value of 18 over 20 divided by 1 is that same value 18 over 20. We’re going to simplify our fraction by dividing 2 over 2 and again this is the value of 1 so we’re not changing the value of the fraction we’re simply simplifying it, 18 divided by 2 is 9, 20 divided by 2 is 10. And our simplified answer is nine tenths.
This is the information that you should include on the division section of your fraction book. You want to start by writing what the division means, “how many groups of blank are in blank?” In division we represent the first fraction and divide it by the value of the second fraction, “ how many groups of one fourth are in one half?” Then what we’re going to do is we’re going to show one half and we place one fourth units underneath it and we count the number of groups of one-fourth in one half. There are 2 groups of one fourth in one half.
We are going back to our SOLVE problem from the beginning of the lesson. Ms. Sherrill has three fourths of a yard of red material. She needs to cut the material into pieces that are one eighth of a yard in length. How many pieces of material will she have? We S the problem, we study the problem, we underline the question and we completed the sentence this problem is asking me to find the number of pieces of material.
We’re now moving to O, we are going to organize our facts. First we are going to identify the facts. Ms. Sherrill has three fourths of a yard of red material. She needs to cut the material into pieces that are one eighth of a yard in length. The second step is to eliminate the unnecessary facts. Ms. Sherrill has three fourths of a yard of material. We need to know how much material she has, so that fact will be necessary. She needs to cut them into pieces that are one eighth of a yard. We also need to know that so we have no unnecessary facts in this problem. Then we’re going to list the necessary facts. We have three fourths of a yard of material, and we need one eighth yard pieces.
We’re going to line up our plan by choosing an operation or operations. Since we’re cutting material, we’re actually dividing it into pieces we’re going to choose division. Our plan of action will be to divide the total quantity of material by the length of each piece.
We are now moving to our V step and we are going to begin to verify our plan with action by estimating our answer. We know we have three fourths of a yard, and we know each piece needs to be one eighth in length, so our estimate is going to be about 5. We’re now going to carry out our plan. We take our three fourths yard and divide it into pieces that are one eighth in length. We’re going to use the concept of reciprocals, remember we divided by one eighth we were multiplying both the fractions by 8 over 1, 3 times 8 is 24, 4 times 1 is 4, when we divide and simplify we have 6 pieces of material.
To complete our word problem we are going to go back to the E step and examine our results. Does our answer make sense? We had 6 pieces of material and we were looking for the number of pieces of material. So, yes our answer makes sense. Is your answer reasonable? Our estimate was 5 and our answer is 6. So, yes our answer is reasonable because our answer is close to our answer of about 5. Is your answer accurate? We’re going to go back and recheck our work and the answer is yes. Then we are going to complete the E step by writing our answer in a complete sentence. Ms. Sherrill will have 6 pieces of material.
Now we are going to go back and answer the essential questions from the beginning of our lesson. Why is it important to know how to build a model for division of fractions? So we will know what the problems mean. What does a division sentence mean? How many groups of blank are in blank? What are the steps for dividing a fraction by a fraction? We create a reciprocal of the second fraction and then multiply.