Level C Lesson 18
Equivalent Fractions
In Lesson 18 the objective is, the student will explain fraction equivalence and compare fractions.
The skills students should have in order to help them in this lesson include, knowledge of comparing whole numbers and basic understanding of fractions.
We will have three essential questions that will be guiding our lesson. Number 1, what are equivalent fractions? Number 2, how do you know when two fractions are equivalent? And number 3, how can you compare fractions if they are not equivalent?
The SOLVE problem for this lesson is, Mary and Jonathon both bought a new box of erasers at the beginning of school. Mary has used one-half of her erasers, and Jonathon has used one-fifth of his erasers. Who has used more erasers?
We will begin by Studying the Problem. First we want to identify where the question is located within the problem and we will underline the question. Who has used more erasers? Now that we’ve identified the question we want to put this question into our own words in the form of a statement. This problem is asking me to find the person who has used more erasers.
During this lesson we will learn how to recognize equivalent fractions and compare fractions. We will use this knowledge to complete this SOLVE problem at the end of the lesson.
Throughout this lesson students will be working together in cooperative pairs. All students should know their role as either Partner A or Partner B before beginning this lesson.
In this lesson we are going to talk about Equivalent Fractions. Fractions are ways to represent parts of whole numbers. In this lesson we will be using fraction strips and number lines to learn about equivalent fractions. Each student will need to have their fraction kit. And each pair of students will need to share colored pencils to participate in the activities in this lesson. Each student pair will also need several beans. We are going to start by modeling several fractions using our fraction strips.
Let’s begin by modeling two-thirds. We can model two-thirds with two of our one-third green fraction strips. Underneath our model of two-thirds let’s model the fraction four-sixths. We can model the fraction four-sixths using four of our one- sixth orange fractions strips. Are the fractions the same size? Yes. Even though the fractions are written differently, they can still be the same size. Two-thirds is equal to four-sixths. The fractions are equivalent fractions.
Let’s record a pictorial model of this example. We will begin by modeling two-thirds using the fraction strip that is provided. The entire fraction strip is equivalent to 1 whole. We need to break up the one whole fraction strip into thirds. Once the fraction strip has been broken into thirds we will shade two-thirds to represent our first fraction. We shade these thirds in green because green is the color of the fraction strip that we use for thirds. Now let’s model four-sixths on the second fraction strip. Again, the entire fraction strip is equivalent to one whole. We will break up the second fraction strip into sixths. And we will shade four of these six to represent the fraction four-sixths. We use orange to shade this fraction as sixths in our fraction kit are orange. The fractions are equivalent because the shading of each fraction covers the same amount of the whole unit strip.
Let’s complete another example together. Going back to our fraction strips we want to model the fraction four-fourths, using the fourth fraction strips. Four-fourths will be four of our yellow one-fourth fraction strips. Next let’s represent the fraction three-thirds, using the third fraction strips. Three-thirds can be modeled with three of our one-third green fraction strips. Are the fractions the same size? Yes. Even though the fractions are written differently, they can still be the same size. Three-thirds is equal to four-fourths. The fractions are equivalent.
Let’s model this example pictorially. We have two fraction bars that are provided for us. Each of these fraction bars is equivalent to one whole. Our first fraction bar will be used to represent four-fourths. We need to break this fraction bar up into four sections, each one representing one-fourth of the whole. Because our fraction is four-fourths we will shade in all four sections using our yellow colored pencil. Now let’s represent three-thirds on the second fraction bar. We want to break the second fraction bar up into thirds. Since our fraction is three-thirds we will shade in all three of these sections. We will use green to shade in the sections as our thirds are represented in green. The fractions are equivalent because the shading of each fraction covers the same amount of the whole unit strip.
In this example, we will be using a number line to determine if fractions are equivalent. Our first fraction is three-fourths and we will model this using our first number line. We need to divide our first number line into fourths. We will have four sections between zero and one. Next we need to label our number line, one-fourth, two-fourths, three-fourths, and one whole would be equivalent to four-fourths. We will place a bean on three-fourths to represent our first fraction.
On our second number line we need to be able to represent six-eighths. So we need to divide this number line into eight sections between zero and one. We want to label each of these sections, one-eighth, two-eights, three-eighths, four-eighths, five-eighths, six-eighths, seven-eighths, and one whole, which is also equivalent to eight-eighths. We will place a bean on six-eighths to represent the second fraction.
Are the fractions three-fourths and six-eighths equivalent to each other? Even though they are divided into a different number of equal sections, the beans are positioned at the same point between zero and one, so they are equivalent. Let’s replace the beans on both of our number lines with a dot using a colored pencil. Are these fractions still the same size? Again, even though they are divided into a different number of equal sections, the dots are positioned at the same point between zero and one, so they are equivalent.
Let’s take a look at another example again using our number line to help us to find if these fractions are equivalent to each other. Our first fraction is two-halves. We will use our first number line to model this fraction. Since our fraction is halves we need two sections between zero and one. We will place one mark at the center of our number line to represent these two sections, and label that mark one-half. Two halves is equivalent to one whole so we will place our bean at one whole.
For our second fraction we want to represent five-sixths on the number line. We will need to create six sections between zero and one, and label each of these sections; one-sixth, two-sixths, three-sixths, four-sixths, five-sixths, and one whole, which is equivalent to six-sixths. We are modeling the fraction five-sixths on this number line, so we will place a bean at five-sixths.
Are the fractions two-halves and five-sixths the same size? Each number line starts are zero and ends at one. The beans are positioned at different points between zero and one, so the fractions are not equivalent. Let’s replace the bean on both number lines with a dot using a colored pencil. Are these fractions the same size? Again, each number line starts at zero and ends at one. The dots are positioned at different points between zero and one, so the fractions are not equivalent.
In this next portion of our lesson we will be creating a pictorial representation to represent fractions, and then writing the relationship between the fractions using the greater than, less than, or equal to symbols.
Let’s begin by modeling with our fraction strips, two different fractions. The first is one-third. We can model one-third using one of our green one-third fraction strips. Our second fraction is two-sixths. We can model two-sixths using two of our orange one-sixth fraction strips. Are the fractions equivalent? Yes, they are equivalent. They are equal to each other.
Let’s create a pictorial representation of each of these fractions. Each fraction bar that has been provided for us is equivalent to one whole unit. One the first fraction bar we will model the fraction two-sixths. We need to break up this fraction bar into six equal sections, and we will shade in two of these sections using our orange colored pencil. On the second fraction bar we will represent one-third. Our second fraction bar is equivalent to one whole, and we will break this fraction bar up into three sections, each representing one-third of the one whole unit strip. We will shade in one of these sections to represent one-third using our green colored pencil. Are these fractions equivalent to each other? Yes, they are equivalent because they both cover the same amount of the whole unit strip. Because the two fractions are equal, you can write the relationship with an equal sign. Two-sixths is equal to one-third.
Let’s take a look at another example together. We will begin by modeling one-half using our fraction strips. We can model one-half using one our brown one-half unit fraction strips. We want to compare one-half to the fraction five-sixths. So we will model five-sixths using our fraction strips. We can model five-sixths using five of our orange one-sixth unit fraction strips. Are these fractions equivalent? No.
Let’s create a pictorial representation of these fractions. We will use our first fraction bar to represent one-half. As the entire fraction bar represents one whole we will split this fraction bar into two sections, and shade one of these sections in brown.
Our second fraction is five-sixths. We will use our second fraction bar which represents one whole and break this one up into six sections. We will shade five of these sections to represent five-sixths of a whole. We will use orange to shade these five sections. We have already recognized that these two fractions are not equivalent to each other. If the fractions are not equivalent, we can use a greater than (>) or less than (<) symbol to show the relationship between the two. Because the first fraction is less than the second fraction we can use the less than < symbol to represent this relationship. One-half is less than (<) five-sixths.
When fractions are not equivalent we can compare them using the less than and greater than symbols, just like when we compare whole numbers.
Let’s model the next two fractions using our fraction strips. We will begin by modeling two-thirds. We will use two green one-third unit fraction strips to model two-thirds. Now let’s model two-sixths. We will use two orange one-sixth unit fraction strips to model two-sixths. What do you notice about the fractions? Both fractions have the same numerator. What fraction is larger? Two-thirds is the larger fraction.
Let’s model these two fractions pictorially. Again, both of our fraction strips are equivalent to one whole. Our first fraction strip will be broken up into thirds to model two-thirds. We will shade in two of these sections in green.
Our second fraction bar will represent two-sixths, so we will need to break this fraction bar up into six sections, and we will shade two of these sections in orange. Which symbol can we place between the two fractions to create a true statement, for two-thirds and two-sixths? Because the first fraction is larger than the second fraction, we can use the greater than (>) symbol, two-thirds > two-sixths.
Let’s take a look at another example together. This time with our fraction strips we want to model one-fourth and two-fourths. We will model one-fourth using one of our yellow one-fourth fraction strips. And we can model two-fourths using two of our yellow one-fourth fraction strips. What do you notice about the fractions? Both fractions have the same denominator in this example. What fraction is larger? Two-fourths is the larger fraction.
Let’s model these two fractions pictorially. Our fraction bars again each represent one whole. We will separate our first fraction bar into four sections to represent fourths, and shade one of these sections in yellow.
Our second fraction bar will also be separated into fourths, and we will shade two sections of our second fraction bar to represent two-fourths. Which symbol can we place between the two fractions to create a true statement? Because the first fraction is less than the second fraction we can use the less than (<) symbol, one-fourth is < two-fourths.
In our next example we want to compare two-fourths and three-fourths. How does this problem differ from problems one and two? This problem uses number lines instead of fraction strips, and the pictorial representation of our fraction strips. We want to represent the fraction two-fourths using the first number line. We need to have four sections between zero and one in order to represent two-fourths. Let’s label each of these sections, one-fourth, two-fourths, three-fourths, and then one whole, which is equivalent to four-fourths. We will place a bean where two-fourths is located on the number line.
We will use the second number line to represent three-fourths. Again, we need to break this number line up into four sections between zero and one. We will label each of the sections as fourths, one-fourth, two-fourths, three-fourths, and one whole, which is equivalent to four-fourths. We will place a bean at three-fourths to represent this fraction on the number line.
We have represented the placement of each fraction on the number line using a bean. Let’s replace our beans with dots to represent these fractions. Which fraction is larger? The fraction three-fourths is larger. What symbol can be used to make a true statement about the relationship between the two fractions? Our first fraction is two-fourths which is the smaller of our two fractions. So we want to use the less than (<) symbol, two-fourths < three-fourths.