Students make sense of multiplication of fractions by using an area model as a way to represent finding a fraction of a hole number and a fraction of a fraction.

Students will:

• draw and shade fractional parts of areas or use square pieces of paper to model fractional parts of a whole and fractional parts of a fraction.
• understand that finding a fraction times a number is the same as finding a fraction of a number.
• learn the meaning of multiplicative inverse.
• see patterns in the results of multiplying with fractions and begin to generalize an algorithm for multiplying fractions without the use of a physical model.
• summarize patterns from examples and problems that relate the resulting value to the fractional and whole numbers involved in the problems.

60 – 90 minutes

• paper
• colored pencils
• index cards with fractions for small groups
• manipulatives: chips, cubes, or dot stickers
• paper to model paper folding fractions
• geoboards or dot paper
• copies of Fraction Cards
• copies of Station Cards
• copies of Assessment Exit Ticket

Use the activities and strategies listed below to meet the needs of your students during the year.

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 below, to reinforce mathematical concepts and check for understanding.

Have a set of fraction cards available. 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.

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 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.

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. The student then can look at where the second and third rubber bands (orange and blue) overlap to determine the product.

This unit is designed to focus on fractions and not on improper fractions or mixed numbers. The goal of this lesson is for students to develop a conceptual understanding of the process of multiplying fractions by finding fractional parts of an area.

“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 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:

                                                * * * * * *

                                                * * * * * *

                                                * * * * * *

                                                * * * * * *

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.”

“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.)

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.

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 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 _____?”

“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 following question: “Look at the different models you created. Can you begin to 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.

To transfer the model of understanding to the area model, on the overhead, chart paper, or white board, draw a rectangle. Explain to students this is the same as the sheet of paper we called a “whole unit.” Take two fractions: ½ and ¼. Break (draw a line through it) the whole into two equal pieces (½) horizontally and shade in 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 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.

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.

• Ongoing teacher observation during small-group work, student interaction, and work stations
• Exit Ticket

• http://www.math.com/school/subject1/lessons/S1U4L4EX.html (walks through multiplication algorithm and gives interactive practice problems)