##
* *Reflection: Discourse and Questioning
Connecting Multiplication and Division - Section 2: Connecting Multiplication and Division

Over the three sets of problems, students were able to see a pattern connecting the multiplication and division problems. They were able to see that dividing a number by 2 is the same as multiplying that number by ½. They were also able to extend this pattern to divisors other than 2. At the end of this section, I posed a question: 10 divided by 'w' will result in the same answer as 10 x ______ . Some students immediately reacted to my use of a variable with “I don’t know!” and “W?! How are we supposed to know what w is?!” I explained that we had been working on finding a connection between multiplication and division, and that I wanted to see if they could use what they’ve learned to demonstrate the connection using variables. I asked if someone had a number we could substitute in for w. One student said we could use 5 for w. 10 divided by 5 = 2, so 10 x _____ will also result in 2? Students participated in a **Think Pair Share** with their partners. Students were able to conclude that 10 x 1/5 would also result in 2. After a bit more discussion, a couple students were able to explain that 10/w is equivalent to 10 x 1/w. I substituted 2 in for w to demonstrate that it indeed worked.

One of my classes had a little extra time, so I asked them what w divided by w would equal. Students were able to say that any number divided by itself will result in 1, so w divided by w will equal 1. Then I asked students what the missing factor would be to also create an answer of 1: w x ______ = 1. Students participated in a **Think Pair Share. **Some students said w, and I explained that w times w would result in w squared. I connected this to multiplication, 3 x 3 is equal to 3 squared, or nine. A student figured it out and I asked her to come to the board to explain her answer. She said w x 1/w would equal one, because if you multiply the two amounts you would get w/w and any number over itself (like 2/2, 3/3, 4/4, etc) equals one.

Some students continue to struggle to extend patterns using variables and this is okay. I think it is important to expose students to these questions and have them work through them. Some students struggle with the idea that a variable could stand for *any *number. The more I push students to extend patterns using variables, the more skilled they will become.

*Extending Patterns Using Variables*

*Discourse and Questioning: Extending Patterns Using Variables*

# Connecting Multiplication and Division

Lesson 12 of 19

## Objective: SWBAT: • Demonstrate the relationship between multiplication and division involving fractions. • Use visual models to divide whole numbers by unit fractions. • Develop strategies for dividing a whole number by a unit fraction.

## Big Idea: How are multiplication and division connected? Why do 11 divided by 2 and 11 times ½ result in the same answer? Students make connections between multiplication and division and work to develop strategies for dividing whole numbers by unit fractions.

*60 minutes*

#### Do Now

*10 min*

See my **Do Now** in my Strategy folder that explains my beginning of class routines.

Often, I create do nows that have problems that connect to the task that students will be working on that day. Today I want students to start thinking about the connection between multiplication and division. Some students may draw a picture of 8 x ½ (4 circles split into halves, or a rectangle model). Other students use the fact that there are 2 halves in a whole, so 8 halves would make 4 wholes.

I ask students to share out their ideas. I then write two more problems on the board: 8 divided by ½ and 8 x 2. I ask students to brainstorm the answers to these problems. Students participate in a **Think Write Pair Share. **I encourage students to draw a picture to show their thinking. I want students to realize that there are 16 halves in 8 wholes. So the answer to both questions is 16. I want students to connect that dividing a number by two has the same result as multiplying by one half.

I ask students, “Why did these problems have the same answer?” I want students to connect that dividing a number by two has the same result as multiplying by one half. Also, I want students to realize that if you multiply two whole numbers (8 x 2), the answer will be larger than either of the factors. When you divide a whole number by a fraction that is less than one, the answer will be larger than either of the factors.

*expand content*

We start Set 1 together. First, we go through and answer the division problems. I call on students to share their ideas. Some students may struggle with answering questions like 7 divided by 2. I ask students to give an estimate. I want students to recognize that the answer should be smaller than 7. Some students may use 8 divided by 2 or 6 divided by 2 to inform their estimate. If students struggle, I draw a picture to show what happens. I can say that if you divide 7 cookies between 2 people, each person will receive 3 ½ cookies.

We then start to connect these problems with multiplication problems. The goal is to create a multiplication problem that involves multiplying the first factor by a number to get the same answer as the matching division problem. For example 8 divided by 2 is 4 and 8 times ½ is 4. We create the multiplication equation to match 7 divided by 2.

Students work to complete the other multiplication equations on their own. When most students are finished, I call on students that participated in a **Think Pair Share **to share out their answers. I ask students to share out what patterns they see **(MP7)**. Students may make connections to what they noticed in the do now problems. I want students to see the relationship between dividing by 2 and multiplying by ½.

We go through similar procedures with sets 2 and 3. I work through the first couple examples together, and then students complete the other problems independently. Then we share out answers and patterns. I want students to recognize that dividing a number by a whole number (6 divided by 6) results in the same answer as multiplying that number by a fraction where that number is a denominator (6 x 1/6). I make sure students can explain why this works. I talk about the problem using cookies as an example if necessary. This is also an opportunity to connect diagrams with the algorithms students have developed regarding fraction multiplication.

*expand content*

#### Painting Tables

*15 min*

I have a volunteer read the information about painting tables. We read problem one and I ask students, “How much paint are you starting with?”, “How much paint does it take to cover one of the small-sized tables?” and “How many small tables could you cover with one gallon of paint?” I show students how I create a model to show how many tables we can cover. See my “Painting Tables Problem 2 Example”. We are going to work on various problems about painting tables that will increase in difficulty over the next few lessons. Students may already know the answer before they create the model and that is fine. I still require them to create the model as a way to support and confirm their answer. I call on students to share out a division number sentence that represents the problem. Then I ask students for a multiplication number sentence that also represents the problem.

Students work on the other problems independently. They are engaging in **MP4: Model with mathematics**. As students work, I walk around and monitor student progress.

If students struggle, I have them return to our first problem and make connections. If students finish 2 and 3, I check in with them quickly to scan their work.

With about 7 minutes remaining, I bring students together to share out their answers and models. Then I ask them to read over the “Looking for Patterns” questions and independently write down their ideas. Students are engaging in **MP7: Look for and make use of structure **and **MP8: Look for and express regularity in repeated reasoning.**

*expand content*

#### Closure and Ticket to Go

*15 min*

For **Closure **I ask students to share out their ideas about the answers to the question 4b involving variables. Students participate in a **Think Pair Share. **I ask students, “Will the answer be larger or smaller than ‘w’?” I call on students to share their ideas. I ask the class if they agree with each student’s ideas and why. Students are engaging with **MP3: Construct viable arguments and critique the reasoning of others**.

I want students to connect these questions to the sets of multiplication and division problems that they worked on earlier in the lesson. When they divided 6 by 1/6 they got 36. You can get the same answer by multiplying 6 by the denominator of the unit fraction, 6 x 6 = 36. This means that if we divide “w” by 1/8, it would be the same as multiplying “w” by 8, or 8w.

I pass out the **Ticket to Go**. Then I pass out the **HW Connecting Multiplication and Division.**

*expand content*

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- UNIT 1: Intro to 6th Grade Math & Number Characteristics
- UNIT 2: The College Project - Working with Decimals
- UNIT 3: Integers and Rational Numbers
- UNIT 4: Fraction Operations
- UNIT 5: Proportional Reasoning: Ratios and Rates
- UNIT 6: Expressions, Equations, & Inequalities
- UNIT 7: Geometry
- UNIT 8: Geometry
- UNIT 9: Statistics
- UNIT 10: Review Unit

- LESSON 1: Pretest
- LESSON 2: Many Names for Fractions
- LESSON 3: What Fraction of the Section Does Each Person Own?
- LESSON 4: Which Fraction is Greater?
- LESSON 5: Adding and Subtracting Fractions Day 1
- LESSON 6: Adding and Subtracting Fractions Day 2
- LESSON 7: Looking for Patterns + Show What You Know
- LESSON 8: Representing Fraction Multiplication
- LESSON 9: Representing Fraction Multiplication Day 2
- LESSON 10: Mixed Number Multiplication
- LESSON 11: The Multiplying Game
- LESSON 12: Connecting Multiplication and Division
- LESSON 13: Dividing Whole Numbers by Fractions
- LESSON 14: Dividing Fractions by Fractions
- LESSON 15: Strategies for Dividing Fractions
- LESSON 16: Strategies for Dividing Fractions Day 2 + Show What You Know
- LESSON 17: Unit Review
- LESSON 18: Unit Closure
- LESSON 19: Unit Test