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: 32/4
For the first task, I wrote the problem in fraction form (32 over 4) in order to expose students to different ways of writing division problems. While solving this task, many students decomposed the 32 into multiples of 4 and then divided by 4: 32:4 = 2(16:4).
Task 2: 64/4
Task 3: 640/4
Then, students solved640:4 = (200x3+40):4. This lead us to a great discussion about the arrays for 640/4 and the patterns: Modeling Arrays for 640:4. I started by modeling a student's strategy (320/4)x2. Then, I said: What if we halved 320? What would be in each of the halves? (160) If we halve the 320 (dividend), what else will we halve? (80... the partial quotient) We continued this conversation with the arrays for 640/2, (80/4)x8, and (40/4)x10. We then discussed: Does anyone see a pattern here? What happens when you decompose 640 into smaller parts? What happens when you decompose 640 into bigger parts? No matter how you decompose 640, do we still get the same quotient each time?
Task 4: 6400/4
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.
Sequence of Division Lessons
When developing a learning progression for multi-digit division, I decided to teach the standard algorithm (long division) prior to developing a conceptual understanding of this process using hands-on tools and visual representations (such as base 10 blocks, money, grid paper, arrays, and ratio tables).
Here's why: In order for students to truly gain a deep understanding of the algorithm, they have to be provided with multiple opportunities to practice the algorithm. If I waited until the end of this division unit to teach this abstract process, students would only have a couple days of practice, leading to very few students achieving proficiency with the standard algorithm.
Also, in order for students to truly see the relationship between division models (such as the money model) and the algorithm, I want to provide them with the opportunity to use the algorithm and models side-by-side in upcoming lessons.
Introduction & Goal
To begin the lesson, I reviewed our current math goal: I can solve division problems with 3-digit and 4-digit dividends using the standard algorithm. Yesterday, we began solving real-world problems involving M&Ms. We discussed the number of M&Ms needed to make three different kinds of treats: The Mint Pretzel Melts, Gingerbread cookies, and M&M cookies. Then, we figured out the number of treats we could make with different amounts of M&Ms. Today, we are going to continue by looking at two more types of treats that can be made with M&Ms.
Music Video Motions
I then explained: Before we begin solving problems, we are going to make a music video! In this music video, I'll need everyone to act out the following dance moves:
1. First you must DIVIDE. (Dividing motion with hands.)
2. Then you MULTIPLY. (Multiplication sign with arms.)
3. And SUB-TRA-CT! (Subtraction sign with one arm.)
4. Then you BRING IT DOWN. (The swim dance move.)
5. The last think you do is CHECK! (Point to head.)
Practice & Dressing Up
We practiced the above dance moves several times. Then, I said: I'll also be needing four volunteers: a dad, mom, sister, and brother. I pulled out costumes (scarf, wigs, sunglasses, hats, gloves, a tie, etc.). Students immediately began giggling! I then pulled glitter sticks (popsicle sticks with student names) to choose volunteers. A couple students didn't want to dress up, while four of the boys were more than willing to partake in the fun!
I then showed the following video and pointed to the Division Poster. We practiced singing the lyrics and acting out the dance moves altogether.
Lights! Camera! Action!
After a couple practice rounds, we recorded this Long Division Music Video. It turned out to be a great way to review the long division steps! It will also be a fun way to review the steps to long division in future lessons!
Long Division Skit
To continue the fun, we then recorded the following Long Division Skit. I asked the student dressed like dad to always divide, the student dressed like mom to always multiply, the student dressed like sister to always subtract, and the student dressed like brother to always bring down. I was hoping this would help all students see the individual steps of long division and how they repeat. This video was memorable and a high-level of student engagement accomplished!
Just as I did with yesterday's lesson, I tried to model complex concepts using simple applications to begin with. This way, students can focus most on making connections and building an understanding of the process of dividing.
For this lesson, students will continue finding the number of treats they are able to make using:
We continued using the Powerpoint Presentation from yesterday: M&M Treats to help students see that division can be applied to everyday problems (Math Practice 4). I began by reviewing our Goal on the first slide. Next, I introduced the next problem: Gratia is making Chocolate M&M Cookies. Each cookie takes _____ M&Ms. If Gratia has _____ M&Ms, how many treats can he make?
I was reminded of how important it is to make learning relevant when Gratia spoke up and said, "That REALLY is my favorite cookie!" I then asked her to come up and count the number of M&Ms in each cookie... She came up to the screen and counted 1...2...3...4...5! Five M&Ms per cookie!
Chocolate M&M Cookies: 16 M&Ms
I then completed the blanks in the problem with students: So how many M&Ms does it take to make one Chocolate M&M Cookie? (5!) If Gratia used the M&Ms on your desk from yesterday to make treats, how many M&Ms does she have? (16!) I wonder how many Chocolate M&M cookies Gratia could make with the 16 M&Ms...
In order to support student understanding of the equal sharing process (division) and to engage students in Math Practice 2 (Reason abstractly & quantitively), I asked a student to model how to divide 16 M&Ms into equal groups of five, 16:5, under the document camera so the whole class could see.
I then said: Let's see if we get the same answer using the standard algorithm! Going on to the next page of the presentation, Blank Algorithm & Verify, I modeled how to use the standard algorithm template to solve 16/5. Next, we verified our solution by multiplying the quotient (3) by the divisor (5). To encourage student input, I said: Wait a minute. The product is 15. Aren't we supposed to get 16? Students responded, "No! We're supposed to add in the remainder! Add 15 + 1 to get 16 M&Ms!" Students also completed the same process on their Algorithm Mats: 16:5.
Chocolate M&M Cookies: 415 M&Ms & 5,062 M&Ms
We then moved on to finding the number of Mint Pretzel Melts we could make with 415 M&Ms (415:5) and 5062 M&Ms (5062:5.). We followed the same process as above. As I solved the problem on the board (Teacher Modeling 415:5 and Teacher Modeling, 5062:5), students competed the same work on their own: Student Work, 415:5 and Student Work, 5062:5.
M&M Trail Mix
To increase the divisor, I showed students the next problem: M&M Trail Mix. Following the same process as above, I Filled in the Blanks for the Trail Mix Problem with: "each cup of trail mix takes 8 M&Ms" and "if we have 16 M&Ms, how many cups could we make?" Again, a volunteer modeled how to divide 16 M&Ms under the document camera: 16:8.
I then modeled (Teacher Modeling, 16:8.) each of the following problems while students solved them on their own algorithm mats:
When we were finished, it was time for students to practice solving the standard algorithm on their own.
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 division problems with 4-digit dividends (without remainders): Division Practice. I created this page by copying 4-digit problems from Math-Aids.com. To increase the assignment expectations from yesterday, I also asked all students to verify their answers by multiplying the quotient and divisor next to each division problem.
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 two examples of conferences with students. You'll notice in the first conference, Student Solving 1260:2, that more space is needed to solve division problems with 4-digit dividends. This student did a beautiful job explaining his steps. I asked a few follow-up questions to make sure he understood what the numbers could represent.
This student also did a great job explaining her steps and the possible word problem that could match the division problem: Student Solving 2733:3.
Most students were able to complete this practice page. Here's a example of a Completed Page.