DIVIDE FRACTIONS
INTRODUCTION
The objective for this lesson on dividing fraction is, the student will interpret and compute quotients of fractions in mathematical and real-world situations.
The skills students should have in order to help them in this lesson include, division of whole numbers, fraction operations and multiplication.
We will have three essential questions that will be guiding our lesson. Number One, why is it important to know how to build a model for division of fractions? Number Two, what does a division sentence mean? And Number Three, what are the steps for dividing a fraction by a fraction?
Begin by completing the warm-up on division problems and practicing wording with division to prepare for the lesson on Dividing Fractions.
SOLVE PROBLEM – INTRODUCTION
The SOLVE problem for this lesson is, Mr. Bryce has four-fifths of a quart of strawberries. He needs to fill one-fourth quart jars. How many jars will he be able to fill?
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 jars will he be able to fill?
Now that we have 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 jars he can fill.
During this lesson we will learn how to divide fractions in order to complete this SOLVE problem at the end of the lesson.
DIVIDING WHOLE NUMBERS BY FRACTIONS – CONCRETE
Two divided by one-fourth
What is the meaning of the division fact two divided by one-fourth? How many groups of one-fourth are in two whole units?
Let’s build a problem with our fractions kits. Let’s start with two whole units.
Identify the divisor. One-fourth
What is the meaning of the divisor? It tells us the size of the items.
Place one-fourth units, label side facing down, under the whole units so that we completely cover both whole units. We place the one-fourth units under each of our whole units.
How many yellow strips, or groups of one-fourth units, are equal to two whole units? Eight
Two divided by two-thirds
What is the meaning of the division fact two divided by two-thirds? How many groups of two-thirds are in two whole units?
Let’s build the problem with our fractions kits.
What do we start with for our model? Two whole units
Identify the divisor. Two-thirds
What does the divisor mean? The divisor tells us how many sections to divide the whole number into.
Do we have a fraction strip that is two-thirds? No
How can we represent two-thirds? Combine two of the one-third fractions strips.
Place groups of two one-third units, label side facing down, under the whole units so that we completely cover both whole units.
How many groups of two one-third units are equal to two whole units? One, two, three
How many groups of two-thirds are in two whole units? One, two, three
There are three groups of two-thirds in two whole units.
DIVIDING FRACTIONS BY FRACTIONS – CONCRETE TO PICTORIAL
One-fourth divided by one-eighth
What is the meaning of the division fact?
How many groups of one-eighth are in one-fourth?
Let’s build the problem with our fractions kits.
What fraction do we represent? One-fourth
What is the divisor? One-eighth
What does the divisor mean? The divisor tells us how many sections to divide the dividend into.
Place one-eighth units, label side facing down, under the one-fourth units.
How many groups of one-eighth units does it take to equal a one-fourth unit? Two
What is the meaning of this division fact? Two-thirds divided by one-ninth
How many groups of one-ninth are in two-thirds?
Let’s build the problem with our fraction kits.
What fraction do we represent? Two-thirds with two one-third units.
What is the divisor? One-ninth
What is the meaning of the divisor? The divisor tells us how many sections to divide the dividend into.
Place one-ninth units, label side facing down, under the one-third units so that they completely cover these units.
How many groups of one-ninth units does it take to equal a two-thirds unit? Six
One-half divided by one-third
What is the meaning of the division fact? How many groups of one-third are in one-half?
Let’s build the problem with our fraction kits.
What fraction do we represent? Lay a one-half unit on your desk.
What is the divisor? One-third
What is the meaning of the divisor? The divisor tells us how many sections to divide the dividend into.
Place one-third units, label side facing down, under the one-half unit so that they completely cover these units.
You can see that two of the green strips show more than one half.
Place two one-sixth units, label side down, under the one-third unit.
One-sixth is half of the green one-third strip, so we need one green strip and another half of a green strip to equal the one-half strip.
How many of the one-third units cover the one-half unit? One and one-half green units completely cover the one-half unit.
How many groups of one-third are in one-half? There are one and one-half groups of thirds in one-half.
What is the division fact? One-half divided by one-third is equal to one and one-half.
RECIPROCALS
Example One: One-half and two
It is not always possible to draw a model of division.
Explain how to write the value of two as a fraction. Two over one
What is the product of one-half and two over one? One
Example Two: One-third and three
What is the value of three as an improper fraction? Three over one
What is the product of one-third and three over one? One
What pattern do you see with the fractions? When you are multiplying two fractions and the second has the numerator and the denominator inverted, the product is one.
Example Three: Three-fourths is a Reciprocal of
Remember that to find the reciprocal of a fraction, we should simply invert the numerator and denominator.
Let’s record the reciprocal of three-fourths
The reciprocal of three-fourths is four thirds
Find the product of three-fourths and four-thirds. Three-fourths times four thirds is equal to one
THE RELATIONSHIP BETWEEN DIVISION AND MULTIPLICATION
One-half divided by one-sixth
Let’s rewrite the problem vertically.
How do we write the reciprocal of one-sixth and multiply to get a denominator of one? Six over one
What process did we use to find equivalent fractions? We multiply the numerator and denominator by the same value.
What does this mean for Problem One? Since we multiplied the denominator by six over one, we will also multiplythe numerator by six over one, which is the reciprocal of one-sixth. Our quotient is three.
SOLVE PROBLEM – COMPLETION
We are now going to go back to the SOLVE problem from the beginning of the lesson. The question was, Mr. Bryce has four-fifths of a quart of strawberries. He needs to fill one-fourth quart jars. How many jars will he be able to fill?
S, we Study the Problem. Underline the question and completed this statement. This problem is asking me to find the number of jars he can fill.
O, Organize the Facts. First we identify the facts. We go back to our original problem and make a strike mark or a vertical line at the end of each fact. Mr. Bryce has four-fifths of a quart of strawberries. / He needs to fill one-fourth quart jars. / How many jars will he be able to fill?
We eliminate any unnecessary facts. We need both of the facts from this problem.
Then we list the necessary facts. There are four-fifths of a quart of strawberries and they are one-quarter quart jars.
L, Line Up a Plan. Write in words what your plan of action will be. Divide the total quantity of strawberries by the amount in each jar.
Choose an operation or operations. Division
Verify Your Plan with Action. First estimate your answer. Our estimate is about three jars.
Carry out your plan. Four-fifths divided by one-fourth is equal to four-fifths times four over one which equals sixteen over five, which equals three and one-fifth jars.
E, Examine Your Results.
Does your answer make sense? Compare your answer to the question. Yes, because we are looking for the number of jars.
Is your answer reasonable? Compare your answer to the estimate. Yes, because it is close to our estimate of about three jars.
Is your answer accurate? Check your work. Yes
Write your answer in a complete sentence. Mr. Bryce can fill three and one-fifth jars of strawberries.
CLOSURE
Now let’s go back and discuss the essential questions from this lesson.
Our first question was, 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 group of blank are in blank?
What are the steps for dividing a fraction by a fraction? Create a reciprocal of the second fraction and then multiply.