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: 6 x 4
For the first task, students decomposed either the 6 or the 4 and created an array to show their thinking: 6 x 4. Others simply made a 6 x 4 array with 24 on the inside. I represented one student's strategy on the board: 6 x 4 Teacher Model.
Task 2: 6 x 14
During the next task, I loved watching this student successfully decompose both factors: 6 x 14. Again, I modeled a student's strategy on the board: 6 x 14 Teacher Model. I wanted students to begin looking for patterns between the arrays.
Task 3: 6 x 214
Then, students solved 6 x 214. After celebrating a student on the last task for experimenting with decomposing more than one factor, many students began trying this too: 6 x 214 and 6 x 214 = 6 x (100+100+10+4). I modeled a couple student strategies on the board: 6 x 214 Teacher Model.
At this point, I asked students to discuss the patterns between each task. One student pointed out that the arrays get bigger with larger factors. Then another student said, "Not necessarily.... it depends on the number of times you decompose." Here are a couple of other patterns students noticed: Student Pattern #1 and Student Pattern #2.
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.
Algorithm & Array Methods
As a side note, my students are already familiar and quite fluent with both the array model and the standard algorithm. 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 countless practice pages on the multiplication algorithm as homework. Consequently, today's lesson will serve as a review and as a foundation for solving more challenging multiplication problems.
Goal & Introduction
To begin, I introduced today's goal: I can use the array model to solve multiplication problems. I explained: During our Number Talk today, we practiced modeling multi-digit multiplication using the array model. Today, we are going to continue using the array model to represent our thinking while multiplying a 1-digit number times a 3-digit number. We are also going to check our answers using the standard multiplication algorithm.
I passed a Multiplication Practice Page to each student. This page can be found at commoncoresheets.com. I also asked students to get out 3 lined sheets of paper. I modeled how to fold and mark each lined paper (front and back) into four rectangles. These sheets of paper will provide students with extra space to represent each multiplication problem on the Multiplication Practice Page using the array method. Here's an example of what one page will look like when students are finished: Grid on Paper.
Modeling the Task
Next, I modeled (Teacher Model #1) and explained: Today, I would like for you to first solve each multiplication problem using the array method. Please take the first lined sheet of paper and place the numbers 1-4 in the top corner of each box. Next to problem number one, I'd like for you to rewrite problem number one from the practice page, 241 x 5. Next, I'd like for you to draw an array. You could draw a horizontal rectangle or a vertical rectangle. On one side, we are going to write the simpler factor, 5. On the other side, we are going to decompose the larger factor by hundreds, tens, and ones. So, for 241, we will decompose this factor into 200 + 40 + 1. Now, let's fill in the products. What is 5 x 200? (Students said: 1000) And what is 5 x 40? (Students said: 200) Finally, what is 5 x 1? (Students said: 5). Perfect, now what do we do? (Students said: Add up all the products.)
After finding the product using the array method, I asked: What could we do to check our work? (Students said: Use the algorithm!) We then solved the algorithm altogether and then moved on to solving the second problem. Following the same process, I modeled this problem as well: Teacher Model #2.
At this point, I wanted students to continue practicing on their own. I went ahead and drew the next two arrays to get students going: Teacher Model.
Math Practice 2: Reason abstractly and quantitatively.
By asking students to solve both the algorithm and the array, I knew students would be engaged in Math Practice 2. Students would have to "make sense of quantities" and "decontextualize an abstract situation (the algorithm) and represent it symbolically (array).
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 loved being able to develop a "game plan" with their partners!
Monitoring Student Understanding
Once students began working, I conferenced with every group. My goal was to support students 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 some examples of student conferences.
This student did a great job explaining both strategies: Student Explaining How She Checks Her Work.
Here, a I try to encourage this student to explain his thinking using more precise math language: Student Explaining 900 x 5.
Almost every conference, I would also encourage students to compare answers: Comparing Answers.
Also, after conferencing with students, I try to always have a Follow Up Conference to make sure students are still on the right track.