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 or using partial products.
Task 1: 83 x 4
For the first task, students were excited to use a newly learned strategy, partial products, alongside of the array strategy: 83 x 4.
Task 2: 166 x 4
During the next task, some students modeled how to double the pervious task using an array. Others decomposed the factors in other ways: 166 x 4. I loved watching this student experiment with fractions: 166 x 4 (fractions)!
Task 3: 2166 x 4
Then, students solved 2166 x 4. In this picture, a student used both partial products and the array method to find the solution.
Task 4: 4322 x 4
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 sit on the front carpet with their white boards. I then reviewed the goal: I can use partial products to solve multiplication problems. I explained: Yesterday, we solved multiplication problems using partial products. Today, we are going to continue working practicing the partial products method, but we are going to try decomposing the factors differently!
Review of Partial Products
I wrote 27 x 3 on the board and asked students to show me how to use partial products to solve this task on their white boards. Here, a student demonstrated her work: 27 x 3. I then asked students to turn an talk: Explain how you used partial products to solve this problem. After giving students time to converse, I asked a student to share and wrote his thinking on the board as he explained, "27 x 3 is the same as 20 x 3 + 7 x 3. The 20 x 3 = 60 and the 7 x 3 = 21. Then, if you add 60 + 21, you'll get 81." I then asked: Which numbers are called partial products? A student responded, "The 60 and the 21." How do you know? "Because 60 and 21 are parts of the whole product. When you add them together, you get the total product."
This is when an amazing and unexpected conversation came about! A student said four of my favorite words during a math lesson, "I see a pattern!" He then came up to the board to explain how 27 x 3 = 9 x 9. Then, another student was inspired by this student's thinking! With some clarification, other students began to see as well: 27 x 3. I was reminded of how important it is to teach students multiple strategies (prime factorization, decomposing, transformation, compensation) to truly develop number sense!
Other Ways to Partition 27 x 3
I explained: I noticed that when we are using partial products, we are decomposing using place value. With 27 x 3, we decomposed 27 into tens (20) and ones (3). Did anyone else notice this too? Students responded with, "Oh yeah... " and "I saw that too!"
To encourage students to investigate 27 x 3 further, I then asked: How else could we decompose the 27? Do we have to always decompose using place value, separating the tens and the ones? Turn and talk about this! In no time, students were excited to share their ideas with others:
I also invited students to share their strategies on the board: 27 x 3 on the Board.
Using Partial Products to Solve 793 x 3
We continued the same process with a more complex problem: 793 x 3. At first, we decomposed the 793 by place value (hundreds, tens, ones) on the board: 793 x 3 = 700 x 3 + 90 x 3 + 3 x 3. Then, students came up with other ways to decompose the 793 instead of just by hundreds, tens, and ones:
Again, I invited students to share their strategies on the board: 793 x 3 on the Board.
Time for Student Practice!
At this point, students were ready to continue practicing with their parters!
Importance of Teaching Partial Products
One of the most important reasons why I teach students how to use partial products in order to solve multiplication problems is to engage students in Math Practice 2 (Reason abstractly and quantitatively). In order to truly understand the connection between place value and multiplication, students must be given opportunities to deconstruct and construct numbers.
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 1 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 paste from several different worksheets found at Math-Aids.com.
I also asked students to staple together three lined sheets of paper. Students divided each page into 4 rectangles. The end result will eventually look like this: Partial Products Page 1. Some students chose to position their page vertically or horizontally.
Next, I Modeled the First Problem to make sure students understood the assignment expectations. I explained: First, I'd like for you to solve the multiplication problem using partial products. I'd like for you to continue experimenting with other ways to decompose the larger factor other than by place value (hundreds, tens, and ones). After you have solved the multiplication problem using partial products, I would like for you to check your work using the algorithm.
After modeling the first problem, I also modeled the Second Problem with less direction, hoping to help students transition from guided practice to independent practice.
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 solving a multiplication problem using partial products and the algorithm: Successfully Using Partial Products.
Here, a I try to encourage this student to use friendlier numbers when decomposing: Choosing Friendly Numbers.
This student did a great job explaining why she solves a multiplication problem using multiple strategies: Checking Work.
Most students were able to complete most multiplication problems using both partial products and the algorithm.
Here's a Completed Algorithm Page and completed partial product pages:
I continue to whole-heartedly believe that conferencing with students is one of the most effective teacher moves ever! Many times, through guided questioning, students come to realize their own mistakes, which is much more effective than the teacher finding the mistakes for students. Other times, just by listening to students as they retrace their steps helps students deepen their understanding as well. Here's an example of a student finding his mistake during a conference: Two Different Answers.
In addition to conferencing, by teaching students to always use two strategies, they are "checking for understanding" on their own!