Today's Number Talk
For a detailed description of the Number Talk procedure, please refer to the Number Talk Explanation. 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: 6342/6
For the first task, some students decomposed 6342 into multiples of 6, such as 6000 + 300 +30 +12, 6342:6 Example A. Others decomposed 6342 in several ways: 6342:6 Example B. After students were done solving, I modeled a new way of subtracting each part from the original dividend: Modeled 6342:6. This strategy seemed to confuse students so I encouraged students to only try it if they'd like!
Task 2: 3171/3
For the next task, I halved both the dividend and the divisor from the first task. Students immediately began guessing what would happen to the quotient, 1057. Some thought 1057 would double while others thought 1057 would be halved. Here's an example of a student who tried multiple strategies: 3171:3 Example A while another student was most comfortable choosing a simpler strategy: 3171:3 Example B. Students exclaimed, "Oh! The quotient stayed the same." Immediately, students tried to figure out why!
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.
Introduction & Goal
To begin the lesson, I invited students to join me on the front carpet. I then explained today's goal: I can solve division problems using the array method. I explained: So far this year, you have done an incredible job representing division problems using the array method when there are no remainders. Today, we are going to experiment with representing division problems with remainders!
I then provided students with a real-world application: Problem. I explained: One thing I haven't told you about my husband, Aaron, is that he LOVES music! He has countless songs on our computer and he's always organizing new playlists. (Students giggled and began commenting on the number of songs they have themselves.) Let's say that Aaron has ___ songs on his computer. He wants to make ____ playlists. If he places the songs evenly in each playlist, how many songs will be in each playlist?
I then filled in the blanks to the problem: What if Aaron has 21 songs and wants to place them into 3 playlists? How can we use the standard algorithm to find the number of songs that would be in each playlist? Students responded, "Divide 21 by 3!" We then solved the 21/3 using long division and continued on to the array: 21songs:3. I modeled on the board while students completed the problems on their white boards: Student, 21:3.
For the array, many students decomposed the 21 into 9+9+3. One student recommended 10+11, but peers took the time to explain that 10 and 11 are not divisible by 3. Next, I modeled how to distribute the "divide by 3" across the equation, (9+9+3)/3 using the distributive property: (9/3) + (9/3) + (3/3). Students caught on quite quickly as we have practiced writing equations for arrays in the past.
Next, we moved on to dividing 22 songs into 3 playlists. Students immediately said, "There's going to be one left over." I wanted to be able to use their prior knowledge of representing 21/3 to solve a problem with a remainder: 22 songs:3. As you can see in the photo, we drew a single square to represent the left over song and labeled it: 1 song left over.
At this point, I wanted to provide students with further practice representing more complex division problems with 3-digit and 4-digit dividends, with and without remainders. So we moved on to solving and discussing the following problems. Instead of modeling first on the board, I allowed students time to solve the problems on their own and then I transitioned to modeling student's thinking on the board.
3-Digit Dividend, No Remainder
3-Digit Dividend, With Remainder
4-Digit Dividend, No Remainder
4-Digit Dividend, With Remainder
At this point, I wanted to provide students with continued practice at their desks.
By teaching students how to connect the standard algorithm with the array representation, I am also engaging students in Math Practice 2: Reason abstractly and quantitatively. Students use the arrays to represent and make sense of the abstract quantities and steps in the standard algorithm!
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 solve multi-digit division problems using the array method so I passed out Array Division Practice created using problems from Math-Aids.com. I had intended to provide students with practice dividing with and without remainders. As it turned out, all of the problems had no remainders! On the bright side, my students appreciated this!
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 are a couple of student conferences. Here's a Student Explaining 18:9. By starting off with easier division problems, students will gain the skills necessary to solve more complex problems.
Here's a Student Explaining 906:6. I was proud of him for catching his own mistake. I also loved watching him add on another 30 to the end of his array. He knew that the partial dividends on the inside did not necessarily have to be ordered from greatest to least.
Most students were able to complete both practice pages. All students solved each division problem using the standard algorithm and the array model. Other students took the time to demonstrate the distributive property as well: Completed Work Page 1 and Completed Work Page 2.
When observing this student conference, Student Explaining 930:2 Part A.MOV and Student Explaining 930:2 Part B.MOV, I am so glad that I chose to begin this unit with long division (even though it was an abstract concept). I am convinced that students need ample practice using this multi-step strategy. If I had waited unit the end of the unit to teach the long division steps, students would not be provided with as much practice.
Also, when students are able to solve the standard algorithm method alongside the array method (representational), they are given the opportunity to develop a conceptual understanding of the long division process.