Today's Number Talk
For a detailed description of the Number Talk procedure, please refer to the Number Talk Explanation.pdf. For this Number Talk, I am encouraging students to represent their thinking using an array model. For each task today, students shared their strategies with peers (sometimes within their group, sometimes with someone across the room). It was great to see students inspiring others to try new methods and it was equally as great to see students examining each other work for possible mistakes!
Task 1: 1240/4
For the first task, some students decomposed the 1240 into 1200 + 40 and then divided both parts by 4. Others solved this task using multiple strategies: 1240:4 Multiple Strategies.jpg. For any students struggling with 1200/4, I would simplify the equation by asking them what 12/4 equals, and then 120/4, followed by 1240/4. Other students decomposed the 1240 and distributed it across the top of the rectangle instead of inside. I would then demonstrate a simpler problem on the board, such as 10 tiles divided into 2 rows equals 5 columns. I then asked: Does it make sense that 10 tiles goes on the outside of the array or on the inside?
Task 2: 2480/4
Throughout every number talk, I continually model student thinking on the board to inspire other students. This also requires students to use math words to explain their thinking instead of relying on a model to represent the math. As students solved each task, I wrote the answers on the board to encourage students to use prior tasks to solve the more complex tasks: Listed Tasks.JPG.
Introduction & Goal
To begin, I invited students to sit on the front carpet with their white boards. I reviewed our current math goal: I can solve multi-digit division problems using partial quotients. I then explained: Today, we are going to practice using two strategies, side-by-side: the standard algorithm (also called long division) and partial quotients. I pointed to the t-chart on the board (Long Divison vs. Partial Quotients) and asked students to create the same chart on their white boards. Here's an example of what we did next: T-Chart.JPG.
To provide students with a real world application (Math Practice 4: Model with Mathematics), I explained Peter's problem and drew the following Pictures.JPG on the board to help students visualize the problem given verbally: Let's say that Peter is having a party for St. Patrick's Day. He has 4 friends that he has invited over to his house for the party. Peter purchases some green M&Ms that he wants to divide up evenly in treat bags for his guests. If he bought ____ M&Ms, what is the greatest number of M&Ms Peter could include in each treat bag?
We then discussed how many bags Peter would need. Some kids said, "Five, including Peter!" Other kids said, "Four, because he has 4 friends coming." We then decided that we needed to calculate for 4 treat bags as the problem didn't specifically state to include Peter.
In order to build a staircase of complexity, I started off by giving students a 2-digit dividend: Let's say that Peter bought 25 M&Ms. What would I divide 25 by to get the greatest number of M&Ms that could be evenly placed in each treat bag? Students said, "4!"
Altogether, we solved 25/4 using the standard algorithm on the left side of the t-chart: Modeled 25:4.JPG. I solved on the front board while students solved on their white boards. I wrote DMSB (from the first division lesson, Shelving Skylanders) off to the side to remind students of the steps. Once we arrived at 6 r 1, I asked: What does 6 represent? What does the 1 represent? Students responded, "The six is the number of M&Ms in each bag. There's one M&M left over." I then asked: What might Peter do with that last M&M? Students giggled and said, "Eat it!"
We then moved on to partial quotients. I explained: Remember, just like a student pointed out yesterday, the partial quotients method is just like repeated subtraction. We're trying to figure out how many groups of 4 there are in 25 so we can just keep subtracting 4 until we either get to 0 or get to a number that is less than 4.
I began by writing "4 x 1" off to the side and subtracting 4 at a time. Students immediately objected, "You don't have to take away one 4 at a time. You could take away more than one group of 4 at a time!" Despite their objections I continued subtracting 4 to prove a point! Students used more efficient ways of subtracting 4 on their own. At the end, I asked, How else could we have subtracted 4? One student, 25:4 = 25 - 2(3x4) +1.JPG, said, "I subtracted three groups of 4 and then another 3 groups of 4." Another student, 25:4 = 25 - (4x5) - (4x1) +1.JPG, said, "I subtracted 5 groups of 4 and then 1 group of 4." I asked: Did we all get the same answer? "Yes!" Why? (I pointed to the six fours on the board.) "Because we all had 6 groups of 4!"
At this point, I provided the next problem, 128/4. I asked students to independently solve the problem using both strategies (the standard algorithm and partial quotients). When students were finished, I asked them to turn and talk about their solutions. During this time, I worked with students one-on-one. Here are a couple examples of student work: 128:4 Student Solution A.JPG and 128:4 Student Solution B.JPG.
Next, we went over the strategies together as a class: Modeled 128:4.JPG.
To provide students with more practice, we solved two more problems, only this time, the problems had 4-digit dividends. Following the same process as above, students solved 3024/4 and 6084/4.
As students finished this problem (Student Example 3042:4.JPG), I asked two students model their thinking on the board:
Standard Algorithm: Solving 3042:4 Using the Algorithm.MOV
Partial Quotients: Solving 3042:4 Using Partial Quotients.mov
For the final problem, I asked students to continue working at their desks. Sometimes when students are gathered at the carpet, it's difficult to maneuver between students. I knew I would be able to equally provide students with if they were sitting at their desks.
Picking math partners is always easy as I already have students placed in desk groups based upon behavior, abilities, and communication skills. Before students began working, I asked them to discuss how they would like to support each other today. I gave them many examples: Do you want to take turns talking out loud? Do you want to solve quietly and then check with each other? Or do you want to turn and talk anytime you get stuck? Students always love being able to develop a "game plan" with their partners!
I wanted to provide students with the opportunity to practice solving real-world division problems so I created and shared the following Google Document: St.PatricksDayParty.doc. My students have their own Google email accounts so they are pretty efficient with copying a shared document and making it their own.
I began by explaining the Problem.png: Peter is assembling party bags for his St. Patrick’s Day Party. He has 9 people coming to his party. He purchases packages of the following items and wants to distribute the items equally. What is the greatest number of each item he could place in each bag? We discussed and highlighted important information: "9 people," "equally," and "greatest number."
Then, we went on to find the number of Pencils.png per bag as a class. I modeled how to solve these multi-step problems by first multiplying off to the side and then we solved 48:9.jpg using the long division and partial quotients methods.
Monitoring Student Understanding
Once students began working, I conferenced with every group. My goal was to support students by providing them with the opportunity to explain their thinking and by asking guiding questions. I also wanted to encourage students to construct viable arguments by using evidence to support their thinking (Math Practice 3).
Here's an example of a conference with a pair of students: Solving 360:9.mov. They both did a great job explaining how to solve the problem and how to to insert images!
Most students were able to complete all the problems but one. I guess I'd rather be over-prepared than under-prepared! Also, when you use Google Docs, wasting paper is not a concern!
Looking back on this lesson, I realized that I should have asked students to solve using the partial quotients method first. By doing this, the rigor of this lesson would have increased as well.
Here's why: Some students would solve a problem like 1962/3 using the standard algorithm and would arrive at the answer, 654. Then, when solving the problem using the partial quotients method, they would use the already found quotient, 653, and take away 600 x 3, 50 x 3, and 4 x 3. This shows that the students were successfully using the place value model, but at the same time, it didn't require as much thought as it would have if they had to solve using the partial quotients method first.