Andrea Palmer PROSPECT HILL ACADEMY CHARTER SCHOOL, SOMERVILLE, MA
6th Grade Math : Unit #2 - The College Project - Working with Decimals : Lesson #6

Dividing with Decimals

Objective: SWBAT: • Make estimates of division problems • Divide whole numbers and decimals • Interpret a remainder
Standards: 6.NS.B.2 6.NS.B.3 MP1 MP6
Subject(s): Math
60 minutes
1 Do Now - 10 minutes

Notes:

  • Before the lesson you will need to copy, cut, and organize the Division Matching cards.  I like to label cards in sets (write #1 on all of first set, etc.) so that students know which cards are theirs.
  • I use the data about division from the pre-test in Unit 1 to Create Homogeneous GroupsFor the do now I have students work in partners or groups of 3.

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.  Here, I want students to connect what is being described in the word problem with a picture.  Students must think about what is going on and what the action will look like. 

After most students have finished most of the matches, I call the class together.  I ask students to share their number sentences and pictures for #4 and #8.  There is a subtle difference in these problems, which I want students to notice. A common mistake is that students confuse the situations that represent 24/4= 6 and 24/6= 4

2 Estimation - 5 minutes

Here I want students to quickly review the vocabulary word quotient and then work on estimating.  I give students a couple minutes to make and write explanations for their estimates.  I stress to students that estimates need to be relatively quick. 

I walk around and monitor student work.  I’m looking to see what strategies students are using.  Some students may round numbers to whole numbers and divide.  Other students may make connections to multiplication.  I am interested to see what students do with #3.  If they struggle I ask, “Do you think the quotient is going to be bigger or smaller than 1.89?  Why?”

I call one student to explain one of their estimates.  These students are students I observed using a particular strategy as I was walking around.  I am not giving exact answers at this time.  We will work on finding exact quotients later in the lesson.  It is important that students are able to use their number sense to make reasonable estimates.

Estimation Reflection
Grappling with Complexity

In this section, I called on a couple students for each problem.  I continued to emphasize that students need to make the number easy to work with using mental math.  Interestingly, a few students in each class said that problem 3 was “impossible”.  I asked them why, and they explained that you can divide a number by something that is larger than that number.  I asked them to round 1.89 to a number that is easier to work with, and they responded with 2.  I told them that they had $2 that they had to share equally between 4 people.  How much money would each person get?  Some students explained that each person would get fifty cents, because fifty cents times four equals $2.  I stated that the problem wasn’t impossible, but that when you divide a number by something larger than that number your quotient will be less than one.  Making the connection to money helped students understand the problem.

3 Practice - 15 minutes

We do number 1 together.  I ask students how they want to find the exact quotient.  I also show them the algorithm.  I stress that students need to use their number sense and the patterns they noticed earlier to decide if their final answer makes sense (MP1)

Students work independently on the rest of the problems.  They can check in with their partner if they are stuck.  I Post A Key so that students can check their work when they finish a page.  I am looking that students are making reasonable estimates and that they are successfully dividing using a strategy of their choice. 

If students successfully complete the practice problems, they can play the Leftovers From 100 game.

If students are struggling, I may intervene in one of the following ways:

  • Ask them what their estimate is and how they got it.
  • Let them use a multiplication chart.
  • Give them a word problem situation that represents the problem.
  • Have them use the grids to organize their problems.
  • Pull a small group of students who are struggling to work together.

 

4 Sharing $$ - 10 minutes

I have students participate in a Think Write Pair Share.  Working with money forces students to interpret the remainder.  I have a few students share and explain their strategies.   Some students will use the picture, others will use the algorithm, others will make connections to multiplication.  Here students are using MP 6: Attend to precision.

We move on to the next page where I ask students to take a couple minutes to analyze the examples.  What do they notice?  What are the similarities? The differences? I give the students a couple minutes to jot down notes.

I want students to see that all of the answers are technically correct, although not as useful as others.  Problem A solves the problem but the answer doesn’t help. I also want students to start moving away from the partial quotient method and towards the standard algorithm.  Problem B turns the remainder into a fraction, but we then need to figure out how to do that. 

A common struggle is that students don’t know where to put a decimal point.  This can be easily remedied if students are able to make reasonable estimates.  For example, offer $67.50 as an answer.  Hopefully students will quickly eliminate this as a possibility, since it is way too big!

I want students to notice the differences between C and D.  In problem C the person ignored the decimal point until the end, and then used estimation to decide where it goes.  Problem D kept the decimal point in the dividend and brought it up to the quotient.  I ask students how could these students check their answers.  I want students to get into the habit of using multiplication to check their answers.

Sharing $$ Reflection
Diverse Entry Points

This problem was a great way to get students to understand dividing with decimals.  Some students who knew the traditional algorithm used it to find their answer of $6.75.  Other students used a traditional method or the partial quotient method to figure out that each person would get 6 ¾ dollars, which is the same as $6.75.  Other students drew pictures to show each person’s money.  There were able to give each person a $5 bill and a $1 bill.  To split up the remaining $3, they traded it in for 12 quarters and each person received 3 quarters.  I had a student who drew a picture present first, followed by students who used the partial product method and the traditional method.  I told students that if someone presented a strategy that they did not use, they needed to draw it on their own paper.  I wanted students to make connections between all of the methods.

5 More Practice - 10 minutes

Just like the previous Practice section, we do number 1 together.  I ask students how they want to find the exact quotient. I stress that students need to use their number sense and estimates to decide if their final answer makes sense. 

Students work independently on the rest of the problems.  They can check in with their partner if they are stuck.  I Post A Key so that students can check their work when they finish a page.  I am looking that students are making reasonable estimates and that they are successfully dividing using a strategy of their choice. 

If students successfully complete the practice problems, they can play the Leftovers From 100 game.

If students are struggling, I may intervene in one of the following ways:

  • Ask them what their estimate is and how they got it.
  • Let them use a multiplication chart.
  • Give them a word problem situation that represents the problem.
  • Have them use the grids to organize their problems.
  • Pull a small group of students who are struggling to work together.

 

Transitioning From the Partial Quotients Method
Student Led Inquiry

I realized that a number of my students were using the partial quotient method for division, instead of the traditional method.  In this section, I modeled how the traditional method and the partial quotients method are connected.

One problem I did was 812.6 divided by 5. Unit 2.5 Transitioning From the Partial Quotients Method.jpgI first had students help me create a reasonable estimate for the problem.  Then  I solved the problem under the document camera, step by step in both ways.  I had students copy down both methods on their paper.  I showed them how the partial quotient method starts to break down when there is a remainder.  I emphasize to students that in sixth grade they need to learn the traditional method, which will help them divide with decimals.  This transition takes time, so I will have to continue to model for students.

6 Closure and Ticket to Go - 10 minutes

I begin the Closure by asking students to look at questions 3 and 4.  What was your estimate? Why?  How did you find the exact quotient.  I look for students who used different strategies and I have them show and explain their work. If there is a common mistake I see students making, I will present it and ask students to address it here.  I am also interested to see what students estimated with #4.  Even if students struggle with estimating, they should be able to see that they answer is going to be smaller than 2.34, since you are dividing it by a number that is larger.

With about five minutes left I pass out the Ticket to Go for students to complete independently.  Then I pass out the HW Dividing with Decimals. I may also assign one of the “Facts about College” or “Working During College” pages for homework, depending on how much students finished.