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.
Task 1: 1 x 2574
Task 2: 2 x 2574
During the next task, students could just double the product to the last task:2 x 2574 (decomposing both factors). Other students decomposed the largest factor: 2 x 2574 (decomposing one factor). I loved watching this Student Challenging Himself by decomposing the 2574 in a creative way.
Task 3: 4 x 2574
Then, students solved 4 x 2574. Again, some students continued experimenting with fractions. Here, one student decomposed the 4 into 2 + 1 + 1/2 + 1/2: 4 x 2574 (using halves). Another student decomposed the 4 into 8 x 1/2: 4 x 2574 (using halves).
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 introduced today's goal: I can use partial products to solve multiplication problems. I explained: Yesterday, we solved 1 digit x 3 digit multiplication problems using the standard algorithm and the array model. Today, we are going to move on to 1 digit x 4 digit multiplication. Also, we are going to be using a new method, called partial products, as well as the standard algorithm to solve these problems. Can you please turn and talk with a nearby student: What do you think partial products are? While students conversed with one another, I silently circled and labeled partial products from our number talk on the board to help students link their understanding of arrays with partial products.
Understanding the Meaning of Partial Products
While students continued their conversations, I used the first number talk task, 1 x 2,574 to begin modeling partial product: Circling Parts. I circled the following partial products inside the array: 2000 + 500 + 70 + 4. I could hear student conversations become richer! Then, I wrote "part" underneath of each partial product. I also labeled these as "partial products." Students began using "part" and "whole" in their conversations. I then asked: What do you think partial products are? One student said, "They are parts of a whole product." Can anyone explain that further? Another student said, "Each product you circled adds up to the whole product. For example, 2000 + 500 + 70 + 4 = 2,574, which is the whole product." I followed this student's suggestion and drew an arrow up to the final product and labeled it "whole product."
To helps students understand this concept on a simpler level, I silently modeled partial sums by adding 25 + 13 on the board: 20 + 10 = 30 and 5 + 3 = 8. I then labeled the 30 and 8 as "parts" that equaled 38, the "whole." Students watched carefully and I was reminded of the power of silently demonstrating a concept. I then asked: What do you think the 30 and 8 could be called? Students knew immediately where I was going with this and responded, "Partial sums!" Do you think there are also partial quotients and partial differences too? "Yes!"
Demonstrating Partial Products
Using the Second Number Talk task, 2 x 2,574, I showed students how to use partial products to solve multiplication problems. I wrote the following on the board:
I lined up each of the digits in the partial products for easy adding. I then asked: What should I do with all of these partial products now? Students responded, "Add them up!" Why should I add them? "Because they are only parts of the whole product." After adding the partial products up as a class, I asked: Why do you think I carefully lined up all of the digits each time we found a partial product? A student said, "To make it easier to add!"
I passed a Multiplication Practice Page from commoncoresheets.com. Just like yesterday, I asked students to get out 3 lined sheets of paper. I modeled how to fold each paper two times in order to get four rectangles. Then, I asked students to draw lines on the folds. Here's a student example: Student Partial Products Page 1.
Modeling the Task
Next, I projected the Multiplication Practice Page and explained: Today, I would like for you to first solve each multiplication problem by finding partial products. Then, I would like for you to check your answer using the standard algorithm. I modeled how to turn the line paper horizontally so that we could use the lines to help line up digits for easy adding. I numbered each of the rectangles 1-4 to represent problem numbers 1-4 on the Multiplication Practice Page.
We began by writing the first problem at the top of the first rectangle: 1,137 x 7. I modeled the first partial product (1000 x 7 = 7000) so that students could see how to use the lines to make their work more precise (Math Practice 6). I wanted students to not only be precise with showing their work, but I also wanted them to accurately calculate their answer. Then, altogether, students helped me identify the remaining partial products: 100 x 7 =700, 30 x 7 = 210, and 7 x 7 = 49. We added them up to get the whole product: 7,959. Here's the final product of this modeling: Teacher Modeling of 1137 x 7.
I then asked: Now, what can do we to make sure the answer is correct? Students knew exactly what I was looking for, "Check your work using the algorithm!" Altogether, we completed the algorithm on the Math Practice Page.
For problem number two, I asked a student to model partial products. I randomly chose a student by pulling a "glitter stick" out of the jar. Here's the Student Modeling 1836 x 4.
At this point, 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!
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 checking his work: Student Checking His Work.
Here, a I try to encourage this student to explain her thinking using more precise math language: Encouraging Precise Math Vocabulary.
Other times, I focused on Providing Some Guidance.
Most students were able to complete at least three pages of partial products practice. When we had ten minutes left, I announced that students could go ahead and complete the rest of the algorithms on the Math Practice Page without showing their work using partial products. Here's a Completed Math Practice Page along with completed partial product pages: Student Partial Products Page 1, Student Partial Products Page 2, and Student Partial Products Page 3.