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 "someone new 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: 38 x 6
Task 2: 38 x 26
During the next task, I asked: How many more 38s than the last task is this? Students responded, "20 more... because 20 + 6 = 26." Students then drew arrays to represent 38 x 26 = (30+4+4) x (20+3+3).
Task 3: 76 x 26
The moment that I wrote the last task on the board, students responded, "I know what you did! You doubled the 38. That means that the answer will be doubled too!" Here, a student decomposed 76 x 26 into (70+6) x (20+6) to arrive at the product, 1,976.
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.
Before starting today's lesson, I wanted to review the standard algorithm to be sure to provide the regular practice needed for students to become proficient using this strategy. I invited students to come up to the front carpet with their whiteboards. One by one, we completed the following problems together. I tried to build the complexity of problems as well as intermix single digit and double digit multplication:
Goal & Introduction
I began today's lesson by introducing the goal: I can solve 2-digit x 2-digit multiplication problems using the in and out box method. I explained: When you are asked to solve multi-digit multiplication problems, I want you to have several strategies to choose from, beyond the traditional algorithm. Today, you will be using the in and out box to solve 2-digit x 2-digit multiplication problems. You will also be using the algorithm to check your work.
To support Math Practice 4: Model with Mathematics, I created a real-world situation in which an in and out box would be an appropriate tool. I began by explaining: Can I tell you my absolute favorite type of candy that I love to purchase any time I go to a candy shop? Peanut butter fudge!
Today, we are going to pretend that there's a candy shop just down the road, owned by Joe. Let's say that Joe has peanut butter fudge for sale. Let me share my Problem with you. I decided to buy some peanut butter fudge from Joe. Each box of peanut butter fudge costs #______ (a certain amount of money). If Mrs. Nelson buys _____ boxes (a certain number of boxes), how much will it cost altogether?
Prior to the lesson, I also created the following Workspace for Modeling. I wanted to make sure we completed all of the following steps:
1. Identify the problem.
2. Construct a rule.
3. Complete the in and out box.
4. Check our work using the algorithm.
For Problem 1, $12 x 6 Boxes, I explained and filled in each of the blanks in the problem template: Let's say that each box of peanut butter fudge costs $12. How much would it cost if I bought 6 boxes? While I modeled the following problem, students also completed each step using their own white boards: Student Work, Problem 1.
1. First, we identified the problem: What multiplication problem would you solve to find the answer to this question? Students responded, "Multiply 12 x 6!"
2. Next, we constructed a rule. I modeled this step to begin with. I wrote "n" and asked: If n equals the number of boxes, what would we multiply by to find the cost of any number of boxes... n times.... Students finished my sentence, "12!"
3. We then completed the in and out box. This is a strategy that we have used in the past. I asked: What should we start with? Students said, "1... 10... and 100!" I always teach students to insert 1, 10, and 100 into the "in" column to begin with. This not only gives students a starting point, but provides students with some perspective.
I then asked: If we buy one box of peanut butter fudge, how much will it cost? ($12) How about 10 boxes? ($120) What if we bought 100 boxes? ($1,200). I then asked: How can we use what we know about 1, 10, and 100 boxes to get closer to 6 boxes? I then showed students how to double the 1 to get 2. I explained: If we double a number on one side, we always double the number on the other side. What is double $12? ($24) If 1 times $12 equals $12, then it makes sense that 2 times $12 equals $24. We continued doubling: If 2 times $12 equals $24, then 4 times $12 will equal $48. Then, we added: (2 x $12) + (4 x $12) = (6 x $12) so $24 + $48 = $72.
4. Finally, we checked the in and out box method using the standard algorithm: Checking 12 x 6.
Math Practice 7
While solving each problem today, I really tried to engage students in Math Practice 7 (Look for and make use of structure) by asking guiding questions, such as "What happens if we double the 2?" As soon as one student excitedly began looking for patterns, others soon joined in!
We then continued on to Problem 2, $12 x 18 Boxes. Again, we used the doubling strategy: If 4 times $12 equals $48, then 8 times $12 equals $96. Then, we added: (10 x $12) + (8 x $12) = (18 x $12) so $120 + $96 = $216.
We then solved Problem 3, $12 x 63 Boxes. Each time I bought more boxes of peanut butter fudge, students gasped as if it really was happening! This time, we figured: If 10 x $12 = $120, then 60 x $12 = $720 (which is $120 x 6). We then found that 3 x $12 = $36. At the end, we added: (60 x $12) + (3 x $12) = (63 x $12) so $720 + $36 = $756.
By the time we got to the last problem, Problem 4, $12 x 97 Boxes, students could hardly wait to see how many boxes of fudge I bought this time! This time, we solved: (100 x $12) - (3 x $12) = (97 x $12) so $1200 - $36 = $1164.
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!
We have a class set of student laptops in our classroom this year. Each student also has a Google email address. Often, I'll create a Google Presentation and share it with students. They will then copy the presentation, making it their own. For this lesson, I created and shared the following presentation: In & Out Box Practice.
Goal & Modeling
Then, I modeled the first problem (Modeled Problem) for the class. I wanted to be sure all students knew and understood the assignment expectations.
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 students conferences. This student shows how ($10 x 25) + ($4 x 25) = ($14 x 25): Student Solving $14 x 25.
This student used her white board to calculate (80 x $53) + (7 x $53) = (87 x $53): Student Solving $53 x 87.
Most students were able to complete most pages of the presentation within this time frame. Here's an example of a finished product: Completed Student Presentation.