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 explore the associative property of addition. I began by explaining the Associative Property Poster: When three or more numbers are added, the sum is the same regardless of the grouping of the addends. If we have (a + b) + c, we could rearrange the addends: a + (b + c) and get the same sum.
Modeling the Associative Property: (2 + 3) + 1 = 6
I wrote this task on the board: Teacher Model (2+3) +1. I reminded students: We always solve whatever is inside the parenthesis first. So what's 2 + 3? Students responded, "Five!" What's 5 + 1? Students responded, "Six!" What would happen if we rearranged the addends? I then wrote: 2 + (3+1) and (2+1) + 3 on the board. Students did the same on their own white boards: Student's White Board (2+3) +1. Students began solving the equations on their own. In no time, I heard expressions of astonishment, "It worked!" "Wow! We go the same answer!" I then responded: I wonder if this happens every time!
Students Testing the Associative Property
Next, I asked students to test the associative property using their own numbers. Here are a few examples:
As students completed this task, I asked them to share their findings with others. It was great hearing some students say, "I don't think the associative property always works. I didn't get the same answer all three times." Others would quickly gather around and respectfully help the student discover a mistake in his/her calculations!
We then shared a few strategies as a class. I modeled students' thinking on the board while the students used math words to explain their thinking. I then asked: Is the associative property always true? Does it always work? Through investigating this property on their own, most students were convinced that they could trust the associative property.
Goal & Introduction
To begin, I invited students to bring their whiteboards and sit on the front carpet, closer to the board. I projected a Smart Notebook lesson: In & Out Box.notebook. If you don't have Smart Notebook Software, this file can be downloaded and opened using: Smart Notebook Express. I introduced today's Goal: I can multiply single-digit x multi-digit numbers using the in & out box method. I explained: So far, we learned how to solve multiplication problems using the array method, standard algorithm, partial products, and compensation. Today, we are going to learn a new strategy called the in and out box.
Starting with What Students Know
I continued on to the next slide in the lesson: Slide 1, 24 x 5. I modeled how to use the In & Out box on the board (Modeled 24 x 5) while students completed the same task on their own whiteboards: Student Whiteboard, 24 x 5. First, we started with a familiar strategy, the standard algorithm. Students practically had this solved before I was able to ask for help with the first step, "What is 5 x 4?"
The In & Out Box: Begin with 1, 10, and 100
We then moved on to the in & out box. I explained: Whenever using the in & out box, you always ask, "Which factor is easiest to multiply by... the 5 or the 24?" Students decided that 5 was a friendlier number the 24. Then, we made the rule together: For this in & out box, we are going to multiply the numbers going "in" by 5. Here's how we represent this rule using a variable: n x 5. Next, I wrote 1, 10, and 100 in the first column under "in." I explained: Whenever using the in and out box, I always like to start by writing 1, 10, and 100 to help give me a little perspective. (Later on, this direction will provide struggling students with a starting place). I then asked: What is 1 x 5? Students responded, "5!" and I wrote the product in the out column. What is 1 x 10? Again, I wrote the product, 50, in the out column. How about 100 x 5? We then wrote 500 in the out column.
Doubling with the In & Out Box
We then reflected on the original problem, 24 x 5. Remember, our target number is 24. That means that we want to try to use the factors we've started with and try to get up to 24. Does anyone see a number we can double to get closer to 24? Students said, "10... We could double the 10 to get to 20. That's close to 24." I then drew an arrow and explained: Here's what's really neat about in and out boxes! If you double a number one side, you can double it's pair on the other side. So if we double the 10, what else can we double? Students caught on quickly, "The 50!" After writing 100 across from 20, I then asked: How many more 5s do we need? We already know what twenty 5s equal. What's our target number again? Students said, "24! That means we need 4 more!" I then said: Okay! So let's just write 4 in the in column! What will the result be in the out column? (20) Now, does anyone see two factors in the "in" column that we could add together to get to 24? Students excitedly raised their hands, "20 + 4!"
I exclaimed: Yes! You're right! I wrote 24 in the "in" column. Here's another rule with the in and out box: If you add two factors together in the "in" column, you have to also add their pairs in the out column. Placing a star next to the 20 and 4, I asked: What was the result of 20 x 5? (100) And what was the product of 4 x 5? (20) So what will we add together? (100 + 20) I modeled how to write 120 across from the 24.
How do know if 120 is the correct product for 24 x 5? Students said, "We got the same answer twice! Once with the algorithm and then with the in & out box!"
At this point, we moved on to the next two slides (Slide 2, 137 x 2 and Slide 3, 784 x 6) and followed the same process. With each new slide, I released more and more responsibility to students by slowly modeling steps (which gave students the opportunity to work ahead of me). Here's what the modeling for these two tasks looked like: Modeled 137 x 2 and Modeled 784 x 6.
Random Number Generator
For the next slides, Slide 4, Random Factors, students took turns clicking on the Random Number Generator which is a fun tool in the Smart Notebook Software that provides students with random numbers. At this point, students were coming up with their own ways of using the in and out box: 359 x 2, Student A and 359 x 2, Student B.
Encouraging Mathematical Discourse
As students finished each task, I asked them to turn and talk about the strategy that they used. This provided students with the opportunity to engage in Math Practice 3: Construct viable arguments and critique the reasoning of others. Many times, conversations helped solidify student understanding of the in and out box method. Other times, students were able to find mistakes by talking through it.
I knew students were ready to continue practicing with their partners!
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!
I passed out Multiplication Practice Page 3 to each student. I wanted students to have a variety of multiplication problems ranging from 1-digit x 2-digit problems to 1-digit x 4-digit problems so I cut and pasted from several different worksheets found at Math-Aids.com. Today, students were only able to solve the first two rows.
Just as we have done in the past, I asked students to staple together two lined sheets of paper. Students divided each page into 4 rectangles: Split the Paper into Four Sections.
Next, I Modeled the First Two Problems to make sure students understood the assignment expectations. I explained: First, I'd like for you to solve the multiplication problem using the in and out box. After you have solved the multiplication problem using the in and out box, I would like for you to check your work using the algorithm.
Monitoring Student Understanding
Once students began working, I conferenced with every group. My goal was to support students by asking guiding questions (listed below).
At first, I provided students with more support and guidance during student conferences: Conferencing with Students.
By the end of our practice time, students were more successful with the in and out box strategy: More Fluent with More Practice.
Again, most students were able to complete the first two rows of multiplication problems using both the in and out box and the algorithm: Student Multiplication Page.
Here are examples of students practicing the in and out box: