Andrea Palmer PROSPECT HILL ACADEMY CHARTER SCHOOL, SOMERVILLE, MA
6th Grade Math : Unit #4 - Fraction Operations : Lesson #13

Dividing Whole Numbers by Fractions

Objective: SWBAT: • Demonstrate the relationship between multiplication and division involving fractions. • Use visual models to divide whole numbers by fractions. • Develop strategies for dividing a whole number by a fraction.
Standards: 6.NS.A.1 MP1 MP2 MP3 MP4 MP6 MP7 MP8
Subject(s): Math
60 minutes
1 Do Now - 5 minutes

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 student to review dividing whole numbers by unit fractions, which we worked on in the previous lesson.  If students struggle, I encourage them to draw 4 circles and see how many sixths or fourths there are.

I ask students to share out their answers.  I also ask students to share what patterns we recognize when we divide a whole number by a unit fraction.  I record their ideas onto a poster and keep it posted so students can use it as a reference.  

2 The Quotient Stays the Same - 10 minutes

Note:

  • I use the data from the ticket to go from Connecting Multiplication and Division to Create Homogeneous GroupsStudents will work in partners.

Students move to sit next to their partners.  I ask student to look at 4 divided by one-sixth equals 24 and identify the quotient, the divisor, and the dividend.  I ask students how we can check our answer to a division problem by multiplying.  I want students to realize that they can multiply the quotient by the divisor and they should get the dividend.  I make students use this vocabulary when they are explaining their answers so they must practice MP6: Attend to precision.  24 times 1/6 equals 4, which serves as a way to check our answer.  Then have students look at the multiplication 4 x _____ = 24.  We need to create a multiplication problem where the answer is the same as the quotient.  I call on a volunteer to share that 4 x 6 = 24.

We go through the 4 divided by ¼ problems together in a similar fashion.  Then students work with their partner to complete the two other rows.  I walk around and monitor student progress.  If students struggle, I encourage them to draw a picture.  If students finish early, I give them other problems to work on like 2 ½ divided by ¼.  I require them to draw a picture to show their thinking and support their answer.

When most partners are finished, we come together to share out answers.  I want students to be comfortable creating a multiplication problem that has an answer that matches the quotient.

3 Painting More Furniture - 10 minutes

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?” and “How much paint does it take to cover one of the small desks?”  I show students how I create a model to show how many tables we can cover.  See my “Unit 4.12 Problem 1 Example”.  We are going to work on various problems about painting tables that will increase in difficulty over the next few lessons.  Students may have an idea of 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 represent the problem.

Students work on problems 2 and 3 with their partners.  They are engaging in MP2: Reason abstractly and quantitatively and 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 the 2 and 3, I check in with them quickly to scan their work. 

4 Looking For Patterns - 20 minutes

We come back together as a class and return to the table on “The Quotient Stays the Same” page.  We use the three blank rows to fill in information about the three painting furniture word problems that students just completed.  I call on students to raise their hand and share out their division number sentences and multiplication number sentences that check their work.  We leave the “Multiplication Problem” columns empty.

I ask students what they notice from looking at the multiplication and division problems for the columns we completed earlier.  Some students may share out that they notice that the division problems have fractions while the multiplication problems do not have fractions (so far).  Other students may notice that the denominator of the fractions in the division problems is the whole number in the multiplication problems.  If no one notices, I ask, “How can we write the whole number factors from the multiplication problems as fractions?”  I explain that in the first multiplication problem, the missing factor was 6.  How could we write six wholes as a fraction?  I want students to share that we can write all of the missing factors as the whole number over 1. 

I present students with their task.  They need to work with their partner to figure out multiplication problems that have the same answer as the quotient for the three painting table questions.  Then they need to complete the “Looking for Patterns” page.

As students work, I walk around and monitor student progress. Students are engaging with MP1: Make sense of problems and persevere in solving them, MP7: Look for and make use of structure and MP8: Look for and express regularity in repeated reasoning.  Students may struggle to create the multiplication problems and that is okay, it is challenging.  I encourage students to guess and check.  For example, 6 x _____ = 9.  If you plug in one, the answer is too small.  If you plug in 2, the answer is too big.  Students will have to guess and check until they figure out that if they multiply 6 x 3/2 or 6 times ½ they will get 9.  Some students may see that 3/2 is the fraction 2/3 flipped.

If students complete this task and their explanation I give them problems like 9 divided by 1 ½ and 1 ½ divided by 1/8 to work on.  I want them to see if their idea about dividing with fractions also works for these problems.

5 Closure and Ticket to Go - 15 minutes

For Closure I ask students what they figured out for the multiplication problems that have the same answer as the quotient.  I have a couple students come to the board to show and explain their thinking.  I want students to see the connection between the fraction divisor and the missing factor to see that they are flipped.

I ask students to share out their ideas about the answers to the question 4b involving variables.  Students participate in a Think Pair ShareI ask students, “Will the answer be larger or smaller than ‘w’? How do you know?  How can you write a multiplication sentence that will result in the same answer?”  I call on students to share their ideas.  I ask the class if they agree with each student’s ideas and why.  Some students may see that dividing w by 2/3 is the same as multiplying w by 3/2 or 1 ½.  I record students’ ideas about dividing a whole number by a fraction on a piece of chart paper so we can revisit them in the next lesson.  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 Dividing Whole Numbers by Fractions.

Connecting Multiplication and Division
Exit Tickets

I collected the tickets to go to see what students dividing whole numbers by fractions.  I corrected the tickets to go and grouped them in the following way:

Novice: Unit 4.12 TTG Novice.jpg

These students struggled to show the connection between multiplication and division.  Also, they struggled to model the division problem in number 2.  This particular student wrote that 3 times 1/6 equals 18 and 3 times ¼ equals 12.  This shows me that she is incorrectly applying a procedure and does not really understand the question.  If 3 times 6 equals 18, than 3 times 1/6 cannot also equal 18.  On problem 2, the model correctly shows the answer of 6, but her number sentences do not match.  I am not sure where the answer of 24 came from.  Perhaps she again incorrectly applied a procedure and multiplied 4 times 3 and then multiplied 12 by 2.  Another common mistake was to solve the problem 4 divided by 1/3 and get 12, rather than solving 4 divided by 2/3. 

This shows me that she is trying to perform shortcuts and lacks understanding of the relationship between multiplication, division, and models.

Approaching Mastery: Unit 4.12 TTG Approaching Mastery.jpg

These students have an adequate understanding of using models and the connection between multiplication and division but they made a careless mistake.  This student solved the problem 4 divided by ¼ rather than 3 divided by ¼.  She drew a model and split 4 wholes into fourths.  

Proficient: Unit 4.12 TTG Proficient.jpg

These students were able to use models and demonstrate the connection between multiplication and division. 

 

For this situation, I did not include an “advanced” category because the content did not require students to explain or analyze their work.   Almost all students were proficient on this ticket to go.  A few students were in the approaching mastery and novice categories.  I will pass these tickets to go back so students can correct their mistakes.  I will also check in with the novice students during the next lesson.