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.
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.
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.
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.
Closure and Ticket to Go
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.