Partial Products & Standard Algorithm Part 1
Lesson 4 of 7
Objective: SWBAT solve 2-digit x 2-digit multiplication problems using the standard algorithm and partial products.
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: 14 x 2
For the first task, many students chose to decompose just the 14: 2(10x4) = 2 (7+7). Here, a student shows three ways to solve 14 x 2 by decomposing the 14 into a 10+4, 7+7, and a 9+5: 14 x 2, Multiple Strategies.
Task 2: 14 x 12
During the next task, I asked: How many more 14s than the last task is this? Students responded, "10 more!" I encouraged them to mentally calculate the math prior to drawing arrays. One student said, "10 x 14 = 140." Another student said, "140 + 28 (the solution to the last task) = 168." Here, a student shows three ways to solve 14 x 12: 14 x 12, Multiple Strategies.
Task 3: 14 x 82
Then, students solved 14 x 82. Again, I asked: How many more 14s than the last task is this? Students responded, "70 more... because 12 + 70 = 82." Then, students used the array method to show 14 x 82. Here's an example of a student who is comfortable decomposing both multiplicands into more than tens and ones: 14 x 82 = (5 + 5+ 4) x (40+40+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: Listed Tasks.
Goal & Introduction
To begin, I invited students to join me on the front carpet with their white boards. I then introduced today's goal: I can solve 2-digit x 2-digit multiplication problems using the standard algorithm and partial products.
I explained: Yesterday, we learned the steps to solve the 2-digit x 2-digit standard algorithm. Today, we are going to continue practicing this method alongside the partial products method. This is important because I want to develop a deeper understanding of how place value connects with multi-digit multiplication.
By providing students with opportunities to practice the standard algorithm and partial products together, I was hoping to engage students in Math Practice 2: Reason abstractly and quantitatively. I knew that this would require students to "attend to the meaning of quantities, not just how to compute them."
Partial Products: 12 x 4
To help connect single digit-multiplication to double-digit multiplication, I first modeled how to use partial products to solve 12 x 4. I reviewed: Remember... whenever we use partial products, we always decompose one or both of the multiplicands. For example, with 12 x 4, we could decompose the 12 into a 10 + 2. Then, we would multiply the 4 x 2 and the 4 x 10 separately. What is 4 x 2? (8) And what is 4 x 10? (40) What do we call the 8 and 40 again? (Partial products!) Why do we call these partial products? (Because they are parts of the full product!)
Partial Products: 12 x 34
Next, we moved on to 12 x 34. I asked: How many more 12s will we have with this problem? Students referred to the previous task and said, "30 more... because 30 + 4 = 34." We then followed the same procedure as above: decomposing the 12 into 10 + 2 and the 34 into a 30 + 4. I then modeled how to multiply 4 x 2 = 8, 4 x 10 =40, 30 x 2 = 60, and 30 x 10 = 300. I will later discover that it is helpful for students to begin with finding the largest partial products first (30 x 10) as it is easier for students to line up digits for adding.
Standard Algorithm: 12 x 34
Next, we moved on to solving the standard algorithm for 12 x 34. We first multiplied the ones: 4 x 2, and then the tens: 4 x 1, to get 48. We then discussed: Doesn't that make sense... that 12 x 4 = 48? We then went on to the next row and I explained: Now that we've multiplied 12 x 4, we need to multiply 12 x the 3... but remember... the 3 isn't really a 3... it's... Students continued my sentence, "3 tens!" So what do I write down first before I begin multiplying? Students responded, "A happy face... or zero!"
To promote higher level thinking, I asked: Why do we write down a zero first? Several students were able to explain, "Because 30 is ten times as much as 3 so you have to multiply the answer by 10." This was a proud moment as it took weeks to encourage students to develop a deeper explanation than "just add a zero." We then continued to find 30 x 12 = 360.
Finally, we added up the partial products to get 408.
As I modeled the partial products and algorithm methods for 12 x 34 students also completed this work on their white boards: 12 x 34 Student Work.
To provide students with further practice, we completed two more problems together: 18 x 25 and 36 x 42. With each problem, I modeled the partial products method and the standard algorithm method. However, with each task, I provided less support and more time for students to work ahead of me. This also gave me the opportunity to work with some students one-on-one.
18 x 25
Here's the modeled methods for 18 x 25: 18 x 25 Partial Products and 18 x 25 Algorithm. Again, students completed the same work on their white boards: 18 x 25 Student Work. As they finished, they turned and talked about their work with peers.
36 x 42
At this point, I wanted students to begin practicing more independently.
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 are lucky enough to have a class set of student laptops 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: Multiplication_ 2 Digit x 2 Digit.
Goal & Modeling
Once students were ready, I reviewed today's Goal on the first page of the presentation. Next, I wanted to model two more problems (pages 2-3 of the presentation) so that I could be sure all students would be successful. Students completed these problems alongside of me. Here are the modeled problems: Modeling 15 x 13 andModeling 25 x 12.
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).
- What is the first step you take when you are using the partial products method?
- What do you do next?
- What do we always need to remember?
- Can you explain your thinking?
- What strategy did you use to help you get the correct answer?
- Did you check with your partner?
- How did you find your mistake?
- Does this always work?
Here are a couple examples of student conferences. First, here is a student Using the Partial Products Method, 26 x 18.
Also, here's an example of a strategy that seemed to help many students: The Covering Up Strategy.
Also, after conferencing with some students who struggled with the algorithm steps, such as this student: Solving the Standard Algorithm, I provided a 1-minute Mid-Lesson Model. It was helpful to color-code the partial products, red: 8 x 12 = 96 and blue: 40 x 12 = 480.
Most students were able to complete most pages of the presentation within this time frame. Here's an example of a finished product: Student Example of Presentation.