Multiplying Fractions by a whole number Using Models

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Objective

Students will multiply a fraction by a whole number using models.

Big Idea

Using a mathematical model, students will be able to visually see the process of how to multiply a fraction by a whole number to find its product in order to gain a deeper understanding to ultimately be able to do it on their own.

Setting The Stage

15 minutes

Fraction Pad

 My students struggle with fractions, so I decide to begin this lesson a little differently.  I want to check and see where students are in their learning, and do a quick review of previously taught skills.  To do this I review vocabulary students need to know in order to access the language we use when working with fractions. 

Factor -  Numbers you can multiply together to get another number.  4 x 5 = 20, 4, and 5 are factors.

Product - The answer you get when two or more numbers are multiplied together.  In the above multiplication sentence, 20 is the product.

Multiple - A number that is the product of a given number and some other number.  20 is a multiple of 4 since 4 x 5=20 (4, 8, 12, 16, 20,...)

It is important that vocabulary is taught in context, so I chose a simple multiplication example that my students should know.

Next, I check to see what students already know about using models to multiply a fraction by a whole number.  I setup the following situation and scenario:

Teacher:

To get my students thinking I write 3/8 on the board.  Then I write 3 X 1/8.  I ask, "Can anyone tell me the relationship between these two? Are they related to each other? How? Is there another way you can represent this?"

Students may not be able to guess any relationship; however, I’m taking them through the steps. 

So I write 1/8 + 1/8 +1/8= 3/8. This way, if students know that 3/8 can be represented by 3 X 1/8 they’ll see the relationship.  If not, I show them the same representation in a simple model. 

When modeling fractions, try to refrain from using circle graphs because it can be hard to represent equal parts.  I ask students if they can show me 3/8 on this rectangle.  Then I ask who can show me 3 x 1/8 using rectangles?  How many rectangles will you need? 3. What are you showing on each one?  1/8 .  Now that we can see these two representations, what can you tell me when you compare them?

 

After the students and I have finished discussing this portion of the lesson, we move into assigned groups to give it a try!

Let's Talk About It

20 minutes

 Students are asked to move to their assigned groups. Using what they have learned so far, they are going to practice solving problems with fractions. Students are allowed to have an open discussion about what they are learning, so that I can understand what they are thinking.

 Materials: paper, pencil, color pencils, and math journals. note taking paper.pdf

Me: "Let’s try another one. Let’s say I have 5/12. Can you represent this as a product of a whole number and a unit fraction?"  I give students time to work it on their own.  If necessary, I go back to our first problem and identify which number was the “whole number” and which was the “unit fraction”.  

Instructions:

Students will use what they know to represent this fraction as a product of a whole number and unit fraction.  

As students are working, I walk around the room and ask how many rectangles they will need?, what fraction will you show on each one?, is there another way you can represent the fraction?  If students are able to show me the two representations, I ask them what can you tell me when you compare them. I give them about 20 minutes or so to work 

 Explanation:

In the fraction 5/12, what number is the unit, and how many of them are there?

Plausible Answers:

The unit is 1/12. And there are 5 of them.

5/12= 5 X 1/12= 1/12 + 1/12 + 1/12 +1/12 + 1/12= 5/12

Remind students:     Again, every fraction can be shown as a sum of a number and unit fractions.

 

Working Together

20 minutes

I ask students to return to their assigned groups.  Now that we have modeled and discussed the concept, I think it would benefit students if they have additional practice working together in their assigned groups.  I want them to have an opportunity to grasp and develop the concepts we just discussed.  I will move into facilitator mode so that I can support and monitor students' progress during this activity.

  I give each group pencils, large poster paper, markers, and rulers.  Because it is essential for them to have the opportunity to apply what they have learned.  I use the same problem used in the modeling stage, however, I give them a different amount to explore. 

Problem:

How much milk is needed for a group of 4 people if each person gets 1/3 cup of milk? How much milk is needed if they each get 2/3 cup of milk?

I ask students to read the given problem and think of ways to express their answers using a mathematical models and multiplication. 

 As students are working, I check for understanding.  For instances, I ask students to discuss how they solved the problem. Some students seem to be having difficulty interpreting the product.  So, I ask them "What are the equal groups?"  I remind them that the number of equal groups is the whole number and the size of each of the equal groups is the fraction.  Then I ask them to write an example just to make sure they understand the given explanation. 

I also explain that each group will have to share their work and explain how they determined their answers with the rest of the class.  This additional practice allows students to work with Math Practice Standard 7: make use of structure.  Sometimes the hands-on visual learning helps my struggling students identify how and why mathematical problems are solved. In this practice the math structure is used to help students understand how to multiply a fraction by a whole number.

 

Students Response:

4x 2/3; each person gets the same amount 2/3 cup, so the total is 2/3 + 2/3 + 2/3 + 2/3= 8/3.  Multiplication can be used to represent repeated addition, so the sum is equal to 4 x 2/3= 8/3= 2 2/3.

I repeat this task with additional questions only if needed.

Independent Closing

20 minutes

Students complete an independent task as a formative assessment.  I ask them to create a mathematical model for problem number 5.  

Student Sample