• Observation during the paper-folding activity can provide information about how well students understand the concept of multiplying fractions.
• Use the Assessment Exit Ticket to evaluate student performance.

Part 1

“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.”

“Let’s review how to multiply whole numbers. When we multiply whole numbers, we can represent the two factors in an array. For instance, if we had the equation
6 × 4 = __, we could represent it like this:

* * * * * *

* * * * * *

* * * * * *

* * * * * *

“How does the array help us to determine the product? What is the product, and 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. Remember, the commutative property of multiplication states that the order of the factors does not affect the product.”

“Now with manipulatives (chips, cubes, dot stickers), create an array to model
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 expression be for 3 × 5?” (5 × 3)

“Does anyone have an idea of how we could visually represent the multiplication of two fractions?” Allow students to dialogue about their ideas and rationale and then 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.)

“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 to show how two fractions can be multiplied.

Explain that the piece of paper represents an area that we will call a whole unit. Use the example . Fold the paper in half horizontally. Open it. Discuss what is represented and shade in . Now fold the paper in fourths vertically. Open it. Discuss what is represented and shade in  using a different color.

Ask questions like the following:

• “What do you notice?”
• “How many pieces do you have in total?”
• “What else do you notice?” (There is an area 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.)
• “What is the product here?” (1/8, because 1 out of 8 sections are shaded.)
• “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.

“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 demonstrated 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.”

As students are working in groups, monitor their performance. Assist students who may not be folding accurately. Visit each group and have students explain their thinking and clarify any misunderstandings.

Sample questions to ask students while they are working:

• “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 do you notice about the relationship of the product to the fractions that produced it?”
•  “What else do you notice?”

The following questions can be used as exploration for students who are showing strong proficiency with the concept.

• “Can you come up with two fractions that will give you the product _____?”
• “Can you come up with an open-ended multiplication sentence to fit your model?” (Answers will vary. Students should choose one fraction and the product:)

Once most students have finished, pose the question: “Look at the different models you created. Can you see a relationship between two fractions and their product?” Give students time to explore and dialogue. Share responses on chart paper and look for patterns of understanding. The goal is that students should begin to see a generalized algorithm for multiplication of fractions.

Part 2

To transfer the model of understanding to the area model, draw a rectangle on the overhead projector, chart paper, or white board. Explain to students this is the same as the sheet of paper we called a “whole unit.” Take two fractions:  and . Break (by drawing a line through it) the whole into two equal pieces () horizontally and shade one of them. Ask students to explain what this represents. Then break the rectangle into four equal pieces vertically and shade in one of them. Ask students to explain what this represents and what they notice. To record the process, have students tell you what two equations are represented by the area model. Do students see the generalized algorithm that was created earlier? Also have students explain how the paper-folding model is similar to the area model.

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 the 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, which follows). 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 (M-5-2-1_Station Cards.docx).

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 (M-5-2-1_Assessment Exit Ticket and KEY.docx).

Extension:

• Routine: Emphasize proper use of vocabulary in lessons and classroom discussions. Allow students to work with partners or in small groups during some activities. Use warm-up or review activities, such as the one that follows, to reinforce mathematical concepts and check for understanding.

Have a set of fraction cards available (M-5-2-1_Fraction Cards.docx). Holding up two cards at random, have students do one of the following activities:

1. Find the product.
2. Write two multiplication sentences along with the product (commutative property).
3. If possible, find two different fractions that will give the same product as the two cards. If it is not possible, explain why.

• Expansion: Introduce students to fractions along with their reciprocals. A reciprocal of a fraction is created by flipping a fraction upside down; the numerator becomes the denominator, and the denominator becomes the numerator. For example, the fraction  has a reciprocal of . The fraction  has a reciprocal of  or 4. A general understanding of improper fractions will be necessary. Another approach is to use the equation along with a pictorial representation. .

Using the array model can show students that . The shaded region shows .

Use the equation .

Since , and   of   is the same as 1 of  and   of  , then 1 of   is , and  of   is . So what’s the sum of   and ? It is 1 whole.

Another way to show this is with a picture.

In the shaded blue striped region, there are 6 pieces. A whole has 6 pieces. So, making .

Through exploration, students can be guided to the understanding that a fraction multiplied by its reciprocal equals one. This is called the multiplicative inverse property. This can be modeled with drawings or manipulatives while explaining several examples.

• Small Group: Based on formative assessments, small-group instruction can be used to help strengthen understanding about multiplication and its effect on fractions. Using an alternate manipulative like geoboards, students can practice modeling multiplication sentences containing fractions. If geoboards are not available or an alternative is preferred, students can complete the activity with dot matrix paper. Remind students that when they folded the paper horizontally and vertically, they got more pieces. The number of pieces was based on the product of the denominators. Model a few paper-folding examples to show students. Using the geoboard with rows of dots, students can determine the area they need by multiplying the denominators of the two fractions.

(; multiplying the denominators 3 and 4, we get 12. Students can create an area showing 12 dots on the geoboard, using two rubber bands to outline the same dimensions as the denominators, 3 rows of dots × 4 rows of dots). Students can then represent the first fraction by looking at the numerator and putting a different color rubber band (blue) on that number of rows. Using a third color of rubber band (orange), students can represent the second fraction by looking at the numerator and putting a rubber band on that number of rows. Students then can look at where the second and third rubber bands (orange and blue) overlap to determine the product.

Make an area with 12 dots using two black rubber bands. Blue represents . Orange represents . There are two dots with both colors so the answer is .

Students then can transfer the process to dot paper or graph paper and identify the product. Having students label the area with fractions and write multiplication sentences can strengthen the algorithm of multiplying fractions using the physical model.

• Technology Connection: Have students create multiplication sentences using fractions and create models of the equations using a spreadsheet program like Microsoft Excel. For instance, the preceding geoboard example can be replicated using cells in a spreadsheet and different fill colors.

An alternative to a spreadsheet program would be a program where students can make a table. Graph paper is another option. Two fractions can be shaded in and the product determined by looking at the overlapping colors and the total number of pieces. Students can create multiple models to represent the same product.