W: [IS.7 - All Students] “We will be using a paper-folding model and an area model to find the product of two fractions. These models will allow you to explore the relationship between two fractions and their product.”
H: “Let’s go back to whole numbers for a moment. When we do multiplication of whole numbers, we can represent the two factors in an array. If we had the equation 6 × 4 = __ , we could represent it like this:
* * * * * *
* * * * * *
* * * * * *
* * * * * * [IS.8 - All Students]
The product would be 24. How does the array help us to determine the product? What does it represent?” (Possible response: We see that there are 6 columns (or groups) and each column has 4 items in it (or 4 rows). So we can see that the total would be 6 + 6 + 6 + 6 = 24 or 6 × 4 = 24.) “If we know that 6 × 4 = 24, then using the commutative property of multiplication we know that 4 × 6 =24. The commutative property of multiplication states that the order of the factors does not affect the product.” [IS.9 - All Students]
“Now with manipulatives (chips, cubes, dot stickers), create an array for
3 × 5 = __. What is the product? What do you notice about the relationship between the factors and the product?” (Do additional problems if necessary.) “Using the commutative property of multiplication, what would the related multiplication equation be for 3 × 5 = __ ?” (5 × 3 = __)
“Does anyone have an idea of how we could visually represent the multiplication of two fractions?” Allow for students to dialogue their ideas and rationale and share them. (Possible responses: Maybe we could start with a whole unit and break it up into pieces. Multiplication means repeated addition for whole numbers, is it the same for fractions? If so, we could use repeated fraction pieces.) [IS.10 - All Students]
“Just as we can visually represent the product of two whole numbers, we can visually represent the product of two fractions. We can do this using a paper-folding model. You will notice something different about the relationship between a product and the factors that produced it when you multiply two fractions and when you multiply two whole numbers.” Use the paper-folding model below to show how two fractions can be multiplied. [IS.11 - All Students]
Explain that the piece of paper represents an area that we will call a whole unit. Use the example ½ × ¼ = __. Fold the paper in half horizontally. Open it. Discuss what is represented and shade in ½. Now fold the paper in fourths vertically. Open it. Discuss what is represented and shade in ¼ using a different color. [IS.12 - All Students]
[IS.13 - Struggling Learners]
[IS.14 - Struggling Learners]
“What do you notice?”
“How many total pieces do you have?”
“What else do you notice?” (Students should notice that there is an area of space that is shaded with both colors.)
“What do you think is represented by the area shaded in by both colors?” (The area shaded by both colors is the product of the two fractions, or a fourth of a half. Since “product” means “multiply,” you have the product of two fractions.)
“Is the relationship between two fractions and their product the same as the relationship between two whole numbers and their product?” (The product of two whole numbers gets larger; the product of two fractions gets smaller.) Do another example if necessary to solidify this concept. [IS.15 - All Students]
E: “I’m going to give each group a set of index cards. On each index card there is a fraction. Each member of the group needs to take two cards and represent the fractions using the paper-folding model (using a regular 8.5 × 11 piece of paper) that I just modeled for you. Once the product is determined, circle or identify in some way the area represented by the product of the two fractions. Then write a multiplication sentence to represent the problem.”
R: As students are working in groups, monitor student performance. Assist students who may not be folding accurately. Visit each group and have students explain their thinking and clarify any misunderstandings. [IS.16 - All Students]
Sample questions to ask students while they are working: [IS.17 - All Students]
“What does the piece of paper represent?” (a whole)
“How do you know how many sections you need?” (look at the denominator)
“How do you know how many sections to shade in?” (look at the numerator)
“What do you notice about the paper and how many parts it has once you start folding?” (number of parts is increasing)
“What do you notice about the size of each piece?” (keeps getting smaller with more folds)
“What else do you notice?”
“What do you notice about the relationship of the product to the fractions that produced it?”
The following questions can be used as exploration for those students who are showing strong proficiency of the concept.
“Can you come up with two fractions that will give you the product _____?”
E: After students have worked as a group and the class has begun to see a generalization for the algorithm of multiplication, have students who demonstrated proficiency during group work rotate to stations, while those students who need further instruction meet as a small group with teacher guidance (see the implementation in the Small Group section below). At each station, alternate between having students find the product of two fractions using the area model (this time drawing the representation rather than folding it) and looking at previously-made area models, then creating a multiplication sentence to show what each model represents (6-1-1_Station Cards.doc).
Encourage students to write the multiplication sentence in two different ways to reinforce the commutative property of multiplication. Have students check their work with answers posted at each station. To bring the lesson to closure, students can be asked the following question: “Were we correct in our thinking when we started to create a generalized algorithm for the multiplication of fractions?” (Possible responses: No. We assumed if we were multiplying we would get a larger product. With fractions, the product of two fractions is smaller than what we start with. Also, we cannot use repeated addition like we can with whole numbers because that also would mean we would have a larger product. It is always important that you think about a whole unit when you are multiplying fractions.)
If further assessment is needed, have students complete the Assessment Exit Ticket ( 6-1-1_Assessment Exit ticket.doc).
