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.
Task 1: 100/10
For the first task, some students showed one strategy for 100:10 = 20x5:10 while others showed Multiple Strategies for 100:10. I also came upon two students Experimenting with Decimals on 100:10. I just loved seeing this!
Task 2: 500/10
During the next task, many students decomposed the 500 into 5 x 100. Here, a student decomposed the 500 into 300 + 200: 500:10 = (300 + 200):10. Also, the two students experimenting with decimals came up with the following solution: Decimals with 500:10.
Task 3: 1000/10
Then, students solved 1000:10. During this whole number talk, I kept encouraging students to write equations to represent their thinking.
Task 4: 3600/10
For the final task, 3600:10, many students used the prior tasks to find the solution. Some students decomposed the 3,600 into 3000 + 600. One student decomposed the 3600 into 2 x 1800: 3600:10 = 1800x2:10.
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.
Algorithm & Array Methods
As a side note, my students are already familiar and quite fluent with using the array model and the standard algorithm to solve multiplication problems. This is because of our daily Number Talks and because I taught students how to use the multiplication algorithm (up to a 1-digit whole number x a 4-digit whole number) earlier in the year. By front-loading this concept, I have been able to send home practice pages on the multiplication algorithm as homework throughout the year. Consequently, today's student practice time will serve as a review and as a foundation for solving more challenging multiplication problems.
Goal & Introduction
I began by introducing today's goal: I can use appropriate tools to solve multiplication problems. I then explained: Today I'm going to give you several problems and I want you to choose appropriate tools to solve each problem.
The goal of today's lesson was to provide students with increasingly complex multiplication problems so that students would be given the opportunity to alter their mathematical tools in order to find an appropriate and efficient way of solving each multiplication problem. I knew that this would be a great way to engage students in Math Practice 5: Use appropriate tools strategically.
Problem One: 3 x 4
Here's your Problem 1: Some students were setting up chairs for a performance. They set up 3 rows. If each row has 4 chairs, how many chairs are there altogether? Please turn and talk: How could we represent this problem? While students discussed, I walked around the room and listened to conversations. Some students said, "We could draw a picture." Other kids said, "We could use tiles." Then, I heard a student share the strategy that I was specifically looking for and I asked him to share with the class. Here's what he said, "We could use chairs to show our thinking." I responded: Okay! Great! Work together to show me your thinking! Here, students rearrange 12 of the chairs in our classroom to represent the problem: Moving Chairs.
Problem Two: 3 x 15
We moved on to Problem 2. Again, I asked students to begin Turning & Talking about Tools before they began. Only this time, students solved the problem individually. Some students used Colored Tiles while others used Pencil & Paper. Here, a student explains how she used Unifix Cubes. This student used the Array model and this student used Playing Cards! We came back together as a class and discussed which tools were appropriate and which tools weren't appropriate for solving this problem. I asked: Why didn't we use chairs to solve this problem? Why were colored tiles or unifix cubes more appropriate tools? Students responded, "We don't have enough chairs." "Also, it's a lot easier to use tiles when the numbers get bigger." I asked: Which numbers? "The factors!" I restated students' ideas: So as the factors get bigger, the appropriateness of math tools change? "Yes!"
Problem Three: 23 x 6
For this problem, we followed the same process. Some students thought that the standard Algorithm was one of the best tools to use. Here, Drawing an Array, a student explains how drawing an array is more practical than using tiles. Others tried using Coins or Using Cards to represent their thinking. As the factors got bigger, students began to find creative ways to make due with the manipulatives they had by having some manipulatives represent more than one. It was great to see students practicing Math Practice 1 and persevering!
Problem Four: 92 x 8
Finally, we moved on to 92 x 8. Here, student explains why she thinks using the 92 x 8 Algorithm as a tool is most appropriate. Other students (especially those who love hands-on manipulatives) took a little more convincing: Money.
Moving On to Student Practice
Now that students had practice choosing appropriate tools to solve multiplication problems, I wanted to move on to our next goal, solving 1 digit x 2 digit multiplication problems.
I passed out a Multiplication Practice page from Commoncoresheets.com for students to practice one digit x two digit multiplication. For the first two rows, I asked students to solve the multiplication problems using an array. For the following rows, I asked students to solve the multiplication problems using the standard algorithm. I wanted students to practice representing one digit x two digit using arrays before moving on to representing one digit x three digit arrays tomorrow. At the same time, I wanted to make sure all students remembered how to use the standard algorithm.
To get students started, I modeled the first two problems as Students Completed the First Two Problems on their own papers.
Conferencing with Students
During this time, I conferenced with students to check for understanding and to ask guiding questions. Here, Solving with the Array, a student explains how to solve 72 x 6 using an array. You'll notice that she says, "6 x 7 is 42 so just add on the 0 to get 6 x 70." In this conference, I try to provide her with a better way of explaining how to use 6 x 7 to get 6 x 70. If you add 0 + 42, it equals 42. Also, when multiplying decimals later on in 5th grade, you don't just "add on a zero."
Other students were mixing up the array representation for division and multiplication: Confusion with Arrays. I checked on these students more frequently until they got the hang of it.
Here's an example of a student Solving with the Algorithm. Again, most of my students are quite fluent with the standard algorithm.
As students finished (Student Example), they checked their answers with one another. Great conversations always result when students defend their thinking!