Today's lesson will begin with a Number Talk. During this opener I will encourage students to represent their thinking using an array model. (For each task today, students share their strategies with peers, sometimes within their group, sometimes with someone across the room.) This is a good lesson with respect to students inspiring others to try new methods. I also find that students pay attention to precision, examining each other work for possible mistakes (MP6)!
Task 1: 24/2
For the first task, students solved 24:2. Many students decomposed the 24 into multiples of 2 (10, 12, 20, 2, 4, etc.)
Task 2: 48/2
During the next task, many students simply doubled the previous task, 48:2. I encouraged students to write an equation to represent their thinking. Some students experimented with decomposing the 48 in a variety of ways: Decomposing 48.
Task 3: 480/2
Task 4: 4800/2
For the final task, some students decomposed 4800 into 4000+800 (4800:2 = (4000+800):2) or into 1000+1000+200+200 (4800:2). I loved watching this student as she represented the idea that 4800/2 = (480/2) x 10 by Multiplying Partial Quotients by 10!
Throughout today's Number Talk, I model examples of student thinking on the board to inspire other students (MP3). In addition to modeling strategies, I use this time to enable my students to use mathematical terms to explain their thinking. I want students to make connections between models and language, rather than relying only on a model to represent their thinking.
As my 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.
After the Number Talk, I introduce today's learning goal:
Yesterday, we learned the steps we take to solve the standard algorithm, which is also called long division. Then, we solved division problems involving 2-digit dividends. Today, we will move on by solving long division problems with 3-digit and 4-digit dividends.
Then, I add a fun motivator. I say, "To help you understand long division better, we are going to use M&Ms today!" Joyful comments always fill the classroom, which creates an atmosphere where students will begin the task feeling very positive about what is to come.
Long Divison Song
Yesterday, students created colorful algorithm mats in order to solve division problems with 2-digit dividends. Today, I handed out a new Up to 4-Digit Dividend Template, specifically designed for problems with 3-digit and 4-digit dividends. Instead of outlining boxes like we did during yesterday's lesson, we simply wrote the letters D, M, S, and B along the side so that we could keep track of the steps in our long division calculation. In order to help students track their progress, I encourage them to use a paper clip to mark their progress.
As we begin the first problem, I label the quotient, the divisor, and the dividend as shown on the Algorithm Mat. And, using the Division Terms poster, we quickly review the meaning of these vocabulary words. Then, I ask students to make sure that they each have 16 candies. I say, "Any extras can be eaten!" (see my reflection on M&Ms and MPs).
Dylan is making Mint Pretzel Melts. Each Mint Pretzel Melt takes _____ M&M. If Dylon has _____ M&Ms, how many treats can he make?
I then Complete the Blanks in the problem with students and discuss the following questions:
In order to help my students visualize the equal sharing process, I ask a student to model how to divide 16 M&Ms into equal groups of one: 16:1 under the document camera so the whole class could see. I then say, "Let's see if we get the same answer using the standard division algorithm!" Before doing so, we discuss the meaning of the Dividend and the Divisor in this task. I'll ask, "Would 16 M&Ms be the dividend or divisor? What number would be the divisor? Can anyone explain why?"
On to the next page of the presentation, Blank Algorithm & Verify, I model how to use the standard algorithm template to solve 16/1: Modeled 16:1. Next, we verified our solution by multiplying the quotient (16) by the divisor (1). Students also completed the same process on their Algorithm Mats: Student Work, 16:1.
I also asked students:
How many M&Ms would Dylon have left over if he used the 16 M&Ms on your desk to make Mint Pretzel Melts?
I expect students will immediately respond, "Zero!" I'll ask, "How do you know?" When they respond, "Because we used all the M&Ms to make the pretzel melts," I'll say, "So what you're saying is that there is a remainder of zero because 0 M&Ms were left over?" And, they will say, "Yes!!" At this point I will say:
Fourth graders, I want you to learn the fancy word we use for the number left over when we are dividing... it's called the remainder.
Please repeat after me: Remainder!
The number left over when we're dividing!
And, I finish this routine by point out our classroom Remainder Poster.
My next move is to dramatically increase the number of M&Ms. First we find the number of Mint Pretzel Melts we could make with 415 M&Ms (415:1). Then, we consider 5062 M&Ms (5062:1). Each time the quantity of M&Ms increases, my students' eyes get bigger! We follow the same process as above. Only this time, I began by asking, "Do you think there will be a remainder... or in other words.... will there be any M&Ms left over?"
For each task, I have my students briefly discussed their thinking. As I solve the problem on the board, students competed the same work on their own: Student Work 415:1 and Student Work 5062:1. Although it may seem surprising, dividing by one using the standard algorithm provides students with the opportunity to make further sense of the process.
When I increase the divisor I also change the context. I show students the Gingerbread Cookies problem. I fill in the blanks of the problem with: "Each cookie takes 2 M&Ms" and "if we have 16 M&Ms, how many treats can we make?" Again, I ask a student volunteer to model how to divide 16 M&Ms under the document camera: 16:2.
I then modeled each of the following problems while students solved them on their own algorithm mats:
Our final context, M&M Cookies, requires us to divide by 3. This time, I have the whole class gather around as we discuss how to divide 16 M&Ms into groups of 3 (see Students Grouping M&Ms). They feel a sense of excitement, since I have established the idea of eating remaining M&Ms earlier in the lesson (see M&M and MP reflection). Then, I model the set of problems below as my students solve each problem at their desks.
During this final set, I encourage my students to work ahead of me as I demonstrate. I keep an eye out for who is tracking me and who is working on their own. This informal assessment helps me to know who to check-in with during the following individual practice segment.
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 today, I plan to ask my students to discuss how they would like to support each other today. I'll give them a few examples:
I find that my student like to have the responsibility of developing a "game plan" with their partner.
For today's practice I want my opportunity to review 2-digit division problems as well as solve 3-digit division problems (without remainders). I created Division Practice by copying both 2-digit and 3-digit problems from Math-Aids.com. For Part 2 of this lesson tomorrow, I'll want students to practice solving 4-digit division problems (without remainders).
By including problems with remainders in today's guided practice, I front-loaded students for tomorrow's lesson. I excluded problems with remainders on the practice page, to ensure student understanding of long division without remainders before moving on to problems with remainders.
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, I conference with a student as she solves and verifies 54/6: Solving 54:6.
In this video, Solving 624:4, a student works through the steps of long division. Understandably, it is easy to get confused when completing this process as students are using every operation besides addition and they are still developing a conceptual understanding of the meaning of division. All students will become more and more comfortable following the long division steps when provided with lots of practice, peer support, and teacher conferencing.
Most students were able to complete this practice page. Here's a example of Completed Work. For today's lesson, I didn't require students to verify their thinking because I didn't want to overwhelm them and I wanted their focus to be on the division algorithm. Tomorrow, I'll ask students to verify as they work through their practice page for on-the-spot confirmation that they are correctly completing the long division problems.