In today's lesson, the students learn to add fractions with unlike denominators. They use a multiplication chart to help them find the least common denominator. Also, they use fraction strips to give them a visual understanding of adding fractions with unlike denominators. This relates to 4.NF.B3a because the students understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
I begin by letting the students know that our lesson for today is adding fractions with unlike denominators. We review the skills from the previous day before getting started on the new skill. I begin by asking a question, "What are like denominators?" I give the students a few minutes to think about the question before answering. Student response: The denominators are the same. One student gives an example. "If you have fractions like 1/5 and 2/5, both of the denominators are the same." I remind the students that we learned yesterday that if you have like denominators, you can go ahead and add or subtract, depending on your problem because the pieces are already the same size. We also learned yesterday, that once you add or subtract, you must write your answer in simplest form. "What is simplest form?" Continually questioning the students assures that the students are listening and paying attention because they do not know who I will call on to answer the question. Also, it allows me to ascertain how much of the information was retained by the students. Student response: by writing the factors out and whatever factor is the same in both fractions, you have to divide by it. There was one thing that the student left out that was very important in simplifying fractions. I call on another student to give me the information. Student response: Do not use the number one. I let the student know that this is correct. I share with the students that even though we are not dividing by the number 1, we are dividing by a fraction that equals 1. I remind the students that in a fraction, when the numerator and the denominator are the same, then that fraction is equivalent to 1. I go on to share with the class that we learned that any number divided by 1 will give you the same number (equivalent).
I share with the students that today, we will add fractions with unlike denominators, this means that the two denominators will be different numbers. When you have to add or subtract fractions with different denominators, you cannot just look at the numerator. You are going to have to find an equivalent fraction. We need to find an equivalent fraction because to add fractions, the denominators must be the same. Because I feel that this lesson is quite comprehensive, I have the students work along with me during whole class discussion time. The powerpoint is on the Smart board. I ask the students to take out a sheet of paper and pencil. I pass out a multiplication chart, along with the fraction strips to each student. As I work on the Smart board, the students work at their desks.
I write the problem 1/4 + 1/3 on the Smart board. I instruct the students to take out the fraction strips for 1/4 and 1/3 and lay them next to each other, as in combining the two (addition). As the students work to solve the problem with paper and pencil, they will use the multiplication chart and fraction strips to get a visual. This will give the students a conceptual understanding of adding fractions with unlike denominators.
I remind students again that we cannot add fractions with different denominators. We need to find a common denominator. On their multiplication sheet, the students look at the multiples of 3 and 4. When you find a common denominator, you have to find a number that is the same in both of your denominators. "What is the first number that you get to that is the same in your 3's and 4's?" Student response: 12. This is called finding the least common denominator. This means that I am trying to find the smallest number that is a multiple for both denominators. There may be other numbers that are the same, but we want to find the smallest number. If you use a larger number, we can still get to the right answer. However, let us try our best to find the smallest number.
On the Smart board, I show the students how to change their fractions into equivalent fractions with 12 as a denominator. I remind them of what we learned during our equivalent fraction lesson. "Multiply the numerator and denominator by the same number." Together, we find the equivalent fraction for 1/4 which is 3/12, and the equivalent fraction for 1/3 is 4/12. I let the students know that at this point, we are doing the exact same thing that we did on yesterday's lesson because now we have common denominators. Together, we add the fractions to get 7/12. To give the students the visual, the students take out their fraction strips for twelfths. They take out 7 of them and line them up below the 1/4 and 1/3. From the picture of Student Work, you can see that the students learned that 1/4 = 3/12 and 1/3 = 4/12 because those amounts are lined under each of those fractions. I tell the students that they do not need fraction stirps to find the answer. This is just to give them a visual of what the math calculations actually mean.
I remind the students to write all of their answers in simplest form. We do this by listing all of the factors that are common in our numerator and denominator. In this problem, 7 and 12. The factors for 7 are 1 and 7. The factors for 12 are 1, 2, 3, 4, 5, and 12. The only number common in both numbers is 1. Therefore, the answer is already written in simplest form.
For this activity, I let the students work in pairs. I give each pair an Add Fractions with Unlike Denominator.docx activity sheet, along with the Multiplication Chart.pdf and fraction strips they already have at their desks (MP5). The students must add fractions with unlike denominators, then write the answers in simplest form.
The fraction strips are used to give the students a visual in order to gain conceptual understanding. Each pair should write the addition problems, then find a common denominator by using the multiplication chart to find the first number common in both denominators. After finding the least common denominator (LCD), the students change the fractions to equivalent fractions using the LCD. At that point, the students can add the fractions. They must write their answers in simplest form by finding a factor that is common in both the numerator and denominator, then dividing to find the answer. The fraction strips are used to justify the answers.
As they work, I monitor and assess their progression of understanding through questioning.
1. What is the first multiple that is common in your two denominators?
2. How can you change them to equivalent fractions?
3. Is your answer in simplest form?
4. What factors are common in both numbers?
As I walk around the classroom, I am questioning the students and looking for common misconceptions among the students. Any misconceptions are addressed at that point, as well as whole class at the end of the activity.
Any groups that finish the assignment early, can go to the computer to practice the skill at the following site until we are ready for the whole group sharing: http://www.mathplayground.com/fractions_add.html
To close the lesson, we review the answers to the problems. This gives those students who still do not understand another opportunity to learn it. I like to use my document camera to show the students' work during this time. Some students do not understand what is being said, but understand clearly when the work is put up for them to see.
I feel that by closing each of my lessons by having students share their work is very important to the success of the lesson. Students need to see good work samples (Student Work - Adding Fractions.jpg), as well as work that may have incorrect information. More than one student may have had the same misconception. During the closing of the lesson, all misconceptions that were spotted during the activity will be addressed whole class.
The students are given an exit ticket (Exit Ticket - Adding Fractions.jpg) to complete at the end of the group activity. It is very important for me to know what each student knows on their own. On the Smart board, the students must write down the following problem to complete: 1/5 + 3/10. I collect all papers from the students as they leave my classroom. The results of the exit ticket: Only 6 students out of 24 could solve for the answer and write it in simplest form. As I stated earlier, I am not surprised because this is a complex skill. One positive thing I found by examining the exit tickets is that quite a few students knew that 10 was the least common denominator. One misconception that I found was that some of the students were trying to multiply both fractions by the same number when changing them to equivalent fractions. For example, because 10 was the common denominator, the equivalent fraction would still be 3/10 because you multiply by 1. Well, some of the students wanted to multiply 1/5 times 1 because they multiplied 3/10 times 1. We will continue the skill on tomorrow with subtracting fractions with unlike denominators.