##
* *Reflection: Modeling
Multiplying Fractions by a whole number Using Models - Section 2: Let's Talk About It

A large percentage of my students did not see the connection between the mathematical model and the problem they were solving. Some students did not use equal parts when they were creating their models, and some students failed to shade the correct amount of parts given away. I wanted them to see that if 6 people were getting 2/3 of a candy bar then it would be the same as

6 x 2/3 or 2/3 + 2/3 + 2/3 + 2/3 + 2/3 + 2/3

Therefore, I think it would have worked better if I would have modeled first on video how to represent the given pieces by using multiplication, addition, and mathematical models. This would have demonstrated how to pay attention to detail.

# Multiplying Fractions by a whole number Using Models

Lesson 14 of 21

## Objective: Students will multiply a fraction by a whole number using models.

#### Setting The Stage

*15 min*

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!

*expand content*

#### Let's Talk About It

*20 min*

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.

#### Resources

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#### Working Together

*20 min*

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.

#### Resources

*expand content*

#### Independent Closing

*20 min*

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

*expand content*

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- LESSON 1: Simplest Form
- LESSON 2: Compare Parts of a Whole
- LESSON 3: Adding and Subtracting Fractions
- LESSON 4: Comparing Fractions
- LESSON 5: Ordering Fractions
- LESSON 6: Solving Problems using Fractions!
- LESSON 7: Modeling Addition of Fractions
- LESSON 8: Improper Fractions and Mixed Numbers
- LESSON 9: Modeling Addition and Subtraction of Mixed Numbers
- LESSON 10: Subtracting Mixed Numbers
- LESSON 11: Decomposing and Composing Mixed Fractions
- LESSON 12: Fractions and Expressions
- LESSON 13: Fractions as Multiples of Unit Fractions: Using Models
- LESSON 14: Multiplying Fractions by a whole number Using Models
- LESSON 15: Decimal Notation VS. Fractions
- LESSON 16: Are They Really The Same?
- LESSON 17: I Would Like to be a Part of the Group!
- LESSON 18: Can I Have a Piece?
- LESSON 19: Whose Piece Is Larger?
- LESSON 20: Not Part, But All Of It
- LESSON 21: Moving from Fractions to Decimals