Partial Products & Standard Algorithm Part 2
Lesson 5 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: 25 x 2
For the first task, students experimented with decomposing one or both multiplicands: 25 x 2.
Task 2: 25 x 12
During the next task, I asked: How many more 25s will we have than the last task? Students said, "10 more because 10 + 2 = 12." I encouraged mental math: What is 10 x 25? Students said, "250... and 250 + 50 = 300." Then, students immediately began modeling this using the array method: 25 x 12. I loved watching some students check their work using the standard algorithm: 25 x 12, Checking with the Algorithm.
Task 3: 25 x 62
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.
Goal & Introduction
To begin, I reviewed the goal from yesterday: I can solve 2-digit x 2-digit multiplication problems using the standard algorithm and partial products.
I explained: Yesterday we practiced using the partial products method and the standard algorithm method to solve 2-digt x 2-digit multiplication problems. Remember, we talked about the importance of developing a deeper understanding of multi-digit multiplication by decomposing numbers based on place value properties. Today, we are going to continue working toward the same goal.
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 the meaning of quantities, not just how to compute them."
Review of Partial Products: 52 x 32
1. After writing 52 x 32 (Teacher Model) on the board, I reviewed: Remember... whenever we use partial products, we always decompose one or both of the multiplicands. For example, with 52 x 32, we could decompose the 52 into a 50 + 2. Then, we could decompose the 32 into a 30 + 2.
2. We would then calculate and record each partial product:
- 2 x 2 = 4
- 2 x 50 = 100
- 30 x 2 = 60
- 30 x 50 = 1500
During this time, I encouraged student engagement my asking:
- What do I do next?
- What does that equal?
- Where do I record the answer?
- What is the answer to a multiplication problem called again?
- Why do we call the 40, 100, 60, and 1500 partial products? (Because they are parts of the full product!)
3. Finally, we add the partial products to find the total product for the problem 52 x 32.
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? Student always love being able to develop a "game plan" with their partners!
I passed out a 2-digit x 2-digit multiplication practice page from Commoncoresheets.com. I wanted students to have multiple opportunities to practice both the algorithm and partial products methods.
Just as we have down in the past, I asked students to staple together two lined sheets of paper. Students divided each page into 4 rectangles. The end result will eventually look like this: Four Rectangles.
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 partial products method within the first rectangle on the lined sheet of paper. After you have solved the multiplication problem using the partial products method, I would like for you to check your work using the algorithm.
Partial Products Order
While modeling the first two problems, a student raised her hand and shared that she likes to begin with finding the largest partial products first (30 x 10 instead of 2 x 5) as it is easier for her to line up digits for adding. Many students tried this student's strategy and agreed that it was easier!
Monitoring Student Understanding
Once students began working, I conferenced with every group. My goal was to support students by asking guiding questions (listed below). I also wanted to encourage students to construct viable arguments by using evidence to support their thinking (Math Practice 3).
- What did you do first?
- Can you explain why you _____?
- What do you see?
- What did you just learn?
- Is this the easiest way to solve this problem?
- Do you see something you could change?
- Is that the same answer as you got over here?
- Why would you use two strategies to solve a problem?
Here are two different students solving the same problem: Solving 47 x 35 and Solving 47 x 35. I think it's interesting how all students approach the same assignment and problems differently. The first student preferred to solve 4 problems at a time using partial products and then he liked being able to go back and solve these four problems using the standard algorithm all at one time. It seemed like the second student wanted to check his answer using the algorithm right away.
Here's another student Solving 77 x 60. The zero in 60 made this problem a little tricky for some students.
Here's an example of the gentle guidance I try to provide students while conferencing with them: Checking Work.
Most students were able to complete all of the multiplication problems using partial products and the algorithm.
Standard Algorithm Practice:
Partial Products Practice: