Compensation

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Objective

SWBAT use compensation to solve multiplication problems.

Big Idea

Students should be able to solve and illustrate multiplication problems using multiple strategies.

Opening

20 minutes

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 partial products.

Task 1: 56 x 8

For the first task, some students used partial products: 56 x 8 = 8(50 +6) and 56 x 8 = 8(20+20+10+3+3). Others used multiple strategies: 56 x 8 = 8 (30+20+6)

 

Task 2: 112 x 4

During the next task, most students used partial products: 112 x 4. I loved how this student caught on to the doubling and halving: 112 x 4 (doubling & halving)

 

Task 3: 224 x 2

Then, students solved 224 x 2. This student showed three different strategies on her board: 224 x 2 (multiple strategies)

 

Task 4: 448 x 1

I wasn't planning on this task, but students will often notice patterns and then they'll raise their hands and say, "I know what the next task is going to be!" One student said, "448 x 1!" I just had to write it up there for a quick discussion. 

 

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.

 

Teacher Demonstration

30 minutes

Goal & Introduction

To begin, I invited students to sit on the front carpet with their white boards. I introduced the goal: I can use compensation to solve multiplication problems. I explained: Yesterday, we solved multiplication problems using partial products. Today, we are going to use compensation. Does anyone remember using the compensation strategy to solve addition problems? Students responded, "Oh yeah!" 

Review: Compensation with Addition

In order to review compensation, I wrote the following on the board: Compensation with Addition (Step 1). I explained: Let's say that we have 36 + 15. To make 36 easier to add, we could add 4, which will turn the 36 into 40. Then we can add the 15. This would equal 55... but is 36 + 15 equal to 55? Students responded, "No....!" So what do we have to do to adjust our answer? Since we "added too much," we have to take away the 4.... and 55 - 4 = 51. As I modeled each step on the board, students also completed the same steps on their white boards: Adding 4 to 36

Next, we reviewed compensation with Addition by solving the same problem as above (36 + 15), only this time, I asked: How else can we use compensation with this problem? A student said, "We can subtract 6!" Here, Compensation with Addition (Step 2), we subtracted six and then adjusted our answer later on by adding six back in: Taking away 6 from 36.

Compensation with Multiplication

We moved on to using compensation when solving multiplication problems. I modeled each problem on the board and students completed each problem on their own white boards as well: 13 x 3 on Student Board. The first task was 13 x 3. I modeled Compensating 13 x 3 (strategy 1) first and then I modeled Compensating 13 x 3 (strategy 2) next. 

We then solved 297 x 3 by Adding to Compensate and Subtracting to Compensate. Again, students solved this on their boards as well: 297 x 3 on Student Board.

To gradually build a staircase of complexity, we then solved 813 x 2

Patterns

I asked students: Can anyone make a conjecture? Did anyone notice a pattern when using compensation to solve multiplication problems? One student said, "If you add, you always adjust later by subtracting. If you subtract, you adjust by adding." I asked: Does everyone agree with this? Should we test this conjecture today? 

Time for Student Practice!

At this point, students were ready to continue practicing with their parters! 

Importance of Teaching Compensation

One of the most important reasons why I teach students how to use compensation 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. 

Student Practice

50 minutes

Choosing 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! 

Getting Started

I passed out Multiplication Practice Page 2 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.

Just as we did yesterday, I 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: Compensation Practice Page 1. Some students chose to position their page vertically or horizontally. 

Next, I Modeled the First Problems to make sure students understood the assignment expectations. I explained: First, I'd like for you to solve the multiplication problem using compensation. After you have solved the multiplication problem using compensation, 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). I also wanted to encourage students to construct viable arguments by using evidence to support their thinking (Math Practice 3). 

  1. What did you do first? 
  2. Can you explain why you _____? 
  3. What did you just learn?
  4. Is this the easiest way to decompose this number? 
  5. Is it just as easy for you to _____ as it is to ______?
  6. Do you see something you could change? 
  7. Is that the same answer as you got over here? 
  8. Why would you use two strategies to solve a problem? 

Student Conferences

This student did a great job solving a multiplication problem using compensation and the algorithm: Student Solving 1509 x 7.

Here, I try to encourage students to engage in mathematical conversations with his partner: Working Together with Partner

Completed 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 compensation practice pages: