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: 3566/10
For the first task, most students decomposed the 3566 into 3000+500+60 and divided each part by 10: 3566:10. I was happy to see that many students also represented their thinking using the standard algorithm and distributive property. As always, I modeled one student's thinking on the board: 3566:10 Modeled.
Task 2: 3566/20
The next task require a bit more thought. It took some students time to figure out 3000/20. Some students decomposed the 3000 into 2000 + 1000 to make it easier. Here's one student's thinking: 3566:20.
Task 3: 3566/100
Then, students solved 3566/100. This turned out to be easier than the last task! Here, a student showed her thinking in multiple ways: 3566:100.
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.
Goal & Introduction
To begin today's lesson I introduced today's goal: I can use the money model to solve division problems involving multi-digit dividends. I explained: Today, you get to represent your thinking with money! I passed out a Bag of Money to each group of about 3 students. Right away, students got the money out and began organizing it! I also asked students to keep their white boards out to help them represent their thinking during the next activity.
I then provided a real-world application to engage students in Math Practice 4: Model with Mathematics. Here's the Problem: There are ____ teachers taking their students on a field trip to Bohart Ski Ranch. They decide to have a fundraiser to help pay for the field trip. If they make $_____ and they split the money evenly, how much will each class get? We just took a ski trip to Bohart last week so students were able to easily connect with the situation!
I passed out a copy of the problem to each group and students placed the Problem in a Page Protector so that the number of teachers and dollars could be interchangeable.
Rolling the Dice
As a fun way to come up with the number of teachers for the first problem, I showed students how to roll a 9-sided die (one-digit divisor). Then, we rolled a pair of regular dice to get the number of dollars raised at the fundraiser (a two-digit dividend). Here's the first problem 2 Teachers, $23 that I rolled. I purposefully manipulated the dice to achieve lower numbers to begin with!
Next, I asked students to work with their groups to divide $23 up evenly. Here, a group discussed what to do: Students Solving $24:3. I enjoyed every minute of this conversation, especially when they began talking about dividing the dollar into quarters! All students did a great job using the money model to represent division. Some students even Checked with the Algorithm..
Once all groups were successful, I celebrated the above group for splitting up the whole dollar into parts. Knowing that we would have more difficult problems ahead of us, such as splitting a remainder of $2 by 3 teachers, I asked student to add "using whole dollars" to the problem in the sheet protector.
Then, I rolled the dice again. Only this time, I rolled 3 regular dice to achieve a 3-digit dividend (amount of money earned at the fundraiser). For this problem, we divided $513 between five teachers: 513:5. With each problem, student explanations became more organized, precise, and clear.
Here, are some Students Dividing up $513 evenly. I was so impressed with this group that I asked the whole class to come and see their work. Altogether, we analyzed their thinking and made positive comments: Analyzing Student Work Together. I was hoping this group might inspire other students!
Finally, I rolled four regular dice to get a 4-digit dividend: 5443:4. Students didn't waste any time dividing up $5443 amongst four teachers. During this process, they had to exchange a $1000 bill for hundreds to get 14 hundred dollar bills. After dividing 14 by 4, students had to exchange 2 hundred dollar bills for 20 tens. Then, they divided 20+40 tens evenly into four groups. Even more exchanging followed! Here's an example of a group who has finally worked through this complex process: Students Dividing up $5443.
I took the opportunity to model the algorithm on the board and discussed how the exchanging for smaller bills correlated with the long division process. For example, we first ask: How many times does 4 go into 5? (1....then subtract: 5 - 4 to get 1... Bring down the 4 to get 14 hundreds... just like the money model!)
At this point, students were ready to continue on with less guidance!
I passed out Dice to each of the groups and explained: For the last part of our math lesson today, you get to roll your own numbers! Today, you will be solving three problems and for each problem, each member of the group will solve using a different strategy. To find the dividend for the first problem (the amount of money raised at the fundraiser), I'd like you to roll a 2-digit number. For the next problem, roll a 3-digit number. For the final problem, roll a 4-digit number. I also wrote Rolling Directions on the board.
Next, I passed out Colored Mats to each group and asked students to label each mat with one of the following division methods. I also explained: Please solve the problem that you roll using each of the following models. Make sure you take turns by rotating the strategies. This way, all the students in your group will get the opportunity to represent their thinking using each method.
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 student conferences from this work time:
2-Digit Dividend: Students Dividing up $52
3-Digit Dividend: Students Solving $666:6
4-Digit Dividend: Students Dividing up $6531
At the end of today's lesson, I was proud of how well students worked together and how students' abilities to explain their thinking developed over the course of the lesson.