I invite students to the carpet and open a discussion about sharing. I tell students of a time when I had to share my favorite sandwich with my sister. How many of you ever had to share your favorite sandwich? Several students raised their hands. Well today we are going to explore fractions a bit more. Are you guys ready? Yes Students are often confused with fractions that are equal to a whole. I notice that most often the biggest misconception is understanding that 1 as the numerator means that only one part of the whole is being represented. This lesson will show students the difference between a fraction represented as a whole compared to other fractional parts.
fraction: a number that shows part of a whole
whole: a complete object or all of the parts to make a complete object
Materials: Apples as Fractions.docx
I display 2 apples on the desk so that students can see. I proceed to cut 1 of the apples in half. What did you notice about what was done to the apple. It was cut. Cut how? In half. How many pieces did I cut the apple into? Two. Is this apple still like the apple on my left? Mixed answers are given. Some students say No, because one is cut and the other is not. Others say that if you put it back together, it will be like the apple on your left. Today we will learn by using models and real objects that as long as no pieces are taken away from the objects, it is still represents a whole. I pick the apple up and put them back together and hold it in one hand while holding the uncut apple in the other hand. Even though I cut this apple into two pieces, it is still a whole apple. On the board, I write that there are two pieces of this apple because it was cut into two parts. Therefore, I can write this as a fraction. I write the number 1 to represent the uncut apple. I write the fraction 2/2 and show students slowly as I put the two pieces together that it makes one whole apple. I tell them that this is why I wrote the fraction 2/2. It shows that I have one whole apple again. This means that, as long as I have all of the pieces needed without removing any of them, I can make any number whole again.
Adding More Parts:
I tell students that I have already cut this apple in half to share with my brother. However, two additional people want a piece of apple also. How many total people are going to be eating this apple? 4. How can I take these two pieces to make them enough to share with two additional people? You can cut it into four pieces. Explain. You can give one person half of yours and your brother can give the other person half of his. Great! I am going to cut this apple to show how it is to be divided equally among four people. I now have four pieces. How many pieces will it take to make this apple whole? 4. This number will be written as your denominator, How many people will be eating the apple? 4. This number will be written as your numerator. I now have the fraction 4/4. This means that there are four pieces of apple and we want to share all four pieces. There's none left over, and even if I get confused, I can put all four pieces back together to see that it represents the same whole apple as the one on my left. I demonstrate for them.
Can anyone tell me what you notice about the fraction 2/2 and 4/4. They have the same number in the numerator and the denominator. Great! Therefore, any time you see a fraction with the same number in the numerator and denominator, it shows that all of the parts of the whole are being used. I show students and that another way to write any fraction that share the same numerator and denominator can be written as 1 because it is the whole.
To move students deeper into the lesson, I ask students to share what they notice so far. I use their responses to check for understanding, and to see if I need to allow them more time to explore fractions.
MP.MP.2. Reason abstractly and quantitatively.
MP. 4. Model with mathematics.
MP. 8. Look for and express regularity in repeated reasoning.
Since all students are doing well with the introduction, I think that they are ready to demonstrate a deeper understanding.
Materials: manila paper, construction paper, scissors, glue, pencils
I give groups of students 3 colored pieces of construction paper, 3 sheets of manila paper, scissors, and glue. You all will cut your manila paper as many times as you like, however, it has to be put back together to represent one whole. When you cut it, count how many pieces you have. That is your whole sheet cut into parts. This number will be your denominator. It has to be glued on this sheet of paper and made whole again. Since these are your parts, what will they be written as? Numerator. Can I get a volunteer to explain what your fractions should look like one you are complete. The same numbers should appear as your numerator and your denominator. Great job! It should be made whole again. I ask students to explain what a fraction is and what a whole number again to make sure that they understand. I ask, what is another way that your fractions should be written? 1 or 1 whole. I give students 5-10 minutes to work their problem. You have to agree on how many pieces your paper will be cut into, write the fraction, glue it together again, and write its representation as a whole number.
I demonstrate the expectation by cutting a sheet of paper into 6 equal parts and gluing it on the construction paper. I write the fraction 6/6 and I also write 1 whole on the board.
I monitor students as they work to make sure that they are grasping the concept. Everyone does well except a few students were struggling with cutting the entire sheets. Therefore, I use pre-printed squares on a sheet of paper for them to use as guides. Even though the lines are pre-printed, they still cut them apart, count the number of parts, write the fraction, and show it written as 1 whole.
I explain to the students that they will work in groups to make a wholesome soup. We will use all whole ingredients such as a whole chicken cut into 8 pieces; therefore the fraction will be 8/8 to represent one whole. Why are we using this fraction? To show that a chicken was cut into 8 pieces and all 8 pieces will be used in the recipe. So, if I cut it into 8's and use all 8 pieces, how many whole chickens am I using? 1. We use 2 carrots cut into fourths; 4 potatoes cut into thirds; two celery stalks cut into eighths; 2 tomatoes cut into halves. Students have colorful pictures of all the ingredients. They use the pictures to cut out the number of pieces they need to make the soup. After students create the equal amount of pieces as directed in the ingredients, they will turn and talk with their neighbor to determine how many whole vegetables were used in the recipe. As students are working, I circle the room to reinforce how to determine how certain fractions represent one whole. I am careful to ask questions that have the students think about how and why the given pieces are equal to 1 whole. I use students responses to determine the level of accuracy, or if students need additional time to explore this concept.
For struggling students, I pass play-doh around to all students that represent the color of the vegetable. The students molds the play doh into the vegetables and cut it into the number of pieces. After the students creates the vegetables, they will be exchanged with other classmates for them to determine how many whole vegetables were used in the recipe.