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: 42/6
For the first task, students decomposed the 42 into multiples of 6: 42:6.
Task 2: 84/6
During the next task, some students used the last task to decompose 84 into 2 x (42x6) while others decomposed the 84 in other ways: 84:6. I liked watching several students check their answer with the algorithm: 84:6 & Algorithm. This shows that students are beginning to feel comfortable using this strategy on their own!
Task 3: 840/6
Then, students solved 840:6 I celebrated students using multiple strategies.
Task 4: 8400/6
For the final task, students decomposed the 8,400 in a variety of ways: 8400:6.
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.
To help teach today's lesson, I created a Powerpoint Presentation: St. Patricks Day Run to help students see that division can be applied to everyday problems (Math Practice 4).
Lesson Introduction & Goal
To begin the lesson, I shared today's Goal: I can solve multi-digit division problems using partial quotients. I explained: Today, we will begin using a new strategy called partial quotients. What does this remind you of? Students responded,"Partial products!" Yes! If partial products refers to "parts of a product," what do you think partial quotients are? "Parts of a quotient!"
I then applied long division to a Real-Life Situation: The Bozeman Runners Club (BRC) is holding a St. Patrick’s Day Run. Each runner will be given a water bottle. The club wants to buy packages of 6 water bottles. How many packages of water will they need to purchase? We then discussed what information is needed to solve the above problem. Students came up with, "We need to know how many runners there are!"
I passed out colored paper inside sheet protectors (green & yellow). I asked students to write "Partial Quotients" at the top using a whiteboard marker. I also asked students to get out their algorithm mats from yesterday's lesson. On one side of their mats, there's a grid for solving long division problems with 2-digit dividends. On the other side of their mats, there's a grid for solving problems with up to 4-digit dividends.
Prior to the lesson, I created a teacher Modeling Chart so that I could model the standard algorithm for division alongside the partial quotients model.
At this point, I introduced Problem 1. Before solving, I asked students: If we have 6 runners, how many bottles of water will we need? "Six!" How many packages of 6 water bottles will we need then? "One!"
We then went on to solve this problem using the algorithm and partial quotients: Teacher Modeling 6:6. After solving the algorithm together, I explained partial quotients: When using the partial quotients model today, we will be subtracting groups of 6 water bottles at a time. Let's begin by taking away one group of 6 water bottles from the total needed. I wrote "x 1" off to the side and subtracted (6 - 6 = 0) while explaining: If we take away 1 group of 6 water bottles, we'll have 0 left. We discussed: How many water bottles will we have left over? Students responded, "Zero, because there's a remainder of 0."
Students also completed both strategies at their desks: Student Work 6:6.
For Problem 2, I wanted to use a visual representation of the water bottles so I copied and pasted more six-packs of water until students were satisfied (Problem 2 Solution): Will one set of 6 water bottles be enough for 15 runners? Students: "No! We need more!" (I copied & pasted another 6-pack.) How about two sets of 6 water bottles? Students: "No! That's only 12!" (I copied & pasted another 6-pack.) How about three sets of 6 water bottles? Students: "Yes! That's 18. We'll have three extra."
By including the water bottle representation, I knew that it would help engage students in Math Practice 2: Reason quantitatively and abstractly. I wanted students to begin looking at the dividend, divisor, quotient, and remainder as more than just numbers!
We then solved this problem using the algorithm and partial quotients: Teacher Modeling 15:6. After solving the algorithm together, I explained partial quotients: Let's begin by taking away one group of 6 water bottles from the total needed. I wrote "x 1" off to the side and subtracted (15 - 6 = 0) while explaining: If we take away 1 group of 6 water bottles, how many water bottles will we still need? Students: "9!" Let's take away another group of 6. (I wrote another "x 1" off to the side.) How many water bottles are needed now? Students: "3!" Can we take away another set of 6 from 3? Students: "No,there's not enough." We discussed: Should we buy another package of 6 water bottles or is it okay to be three water bottles short? Students decided, "We better buy 3 sets of 6 water bottles because it's better to have too many than not enough!"
Again, students completed both strategies at their desks alongside of me: Student Work 15:6.
We continued in the same fashion as above with the next problem: Problem 3. Again, I added sets of 6 water bottles at a time until students felt we had enough: Problem 3 Solution. This was an important step as it helped students see the importance of buying one more package when there was a remainder.
I then modeled the problem for the class: Teacher Modeling 37:6. This time, I asked: Do we still want to take away one six at a time (I began subtracting one six at a time and writing "x 1 off to the side for each 6 I took away) or should we take away more than one six at a time? Students agreed that subtracting one six at a time wasn't a very efficient method so we took away 2 groups of 6 (37-12=25) and I wrote "x 2" off to the side. Then a student suggested that we take away 4 groups 5 (25-24=1) and I wrote "x 4 off to the side." I asked: How many groups of 6 did we take away altogether? "Six!" And what is the remainder? "One!"
Again, students completed both strategies at their desks alongside of me: Student Work 37:6.
At this point, a student raised his hand and said, "I get it! Just like multiplication is repeated addition, division is like repeated subtraction!" This was perfect timing!
I then modeled the problem for the class and students completed the problem at their desks: Student Work 50:6.
We went on to solve Problem 5. With the larger dividends, we skipped solving the problem ahead of time by copying and pasting sets of 6 water bottles. Instead, we immediately began solving the problem using the algorithm and partial quotients.
At first, when using the partial quotients method, I began taking away 1 group of six water bottles at a time: Teacher Modeling 138:6 Part A. This drove students crazy! Students objected: "No! "here's an easier way!" "That will take too long!"
A student then guided me through a more efficient way: Teacher Modeling 138:6 Part B. We first took away 20 groups of 6 (138-120=18 and then 3 groups of 6 (18-18=0).
Again, students completed both strategies at their desks alongside of me: Student Work 138:6.
To provide students with guided practice, we continued on by solving problems 6 and 7. Only for these problems, I provided students with time to solve the problems using the algorithm and partial quotients on their own prior to modeling on the board.
I explained: For the next couple problems, I'd like for you to try solving the problem ahead of me. You can work with a partner or on your own. Either way, I want you to turn and talk about your work as you finish!
For Problem 6, most students began by solving the algorithm first. Then, they moved on to using the partial quotients method.
I prompted students by asking: Who wants to take away one group of 6 at a time? How about 2? How about 10 or 100 groups of 6? Students immediately began taking away groups of 6 as they felt comfortable. During this time, I conferenced with as many students as possible to provide one-on-one guidance.
Here are a few examples of student work:
Here, Student A Work 507:6, did a beautiful job taking away groups of 6. He ended up with three too many in his quotient (the x1, x1, and x1). I think that he was trying to take away three more as there was a remainder of 3.
This student, Student B Work 507:6, took away 6x40, 6x40, and 6x4 and arrived at the quotient: 84 r3.
Another student, Student C Work 507:6, took away 6x10, 6x20, 6x10, and 6x6. It was wonderful to watch students come up with so many ways to subtract 6!
After some time, I modeled a student's strategy on the board: Teacher Modeling 507:6.
For Problem 7, I explained: Let's say even more runners signed up for the BRC marathon! This time, there's over 1000 runners! One student commented, "Mrs. Nelson, we're going to have to have remainders on this one so the people carrying all this water have a couple bottles too!"
Again, students began solving the problem on their own to begin with.
Finally, I modeled a student's thinking on the board: Teacher Modeling 1502:6.