Compensating to Compute Smaller Numbers
Lesson 11 of 16
Objective: SWBAT use compensation to check the addition and subtraction algorithms for accuracy.
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 open number line model.
Task 1: 1,800 + 400
For the first task, students added parts of 400 to 1,800. For example, many students decomposed 400 into 200 + 200. Starting at 1,800 on their number lines, they took a jump of 200 to reach a landmark number: 2,000. Then, they took one more jump of 200.
Task 2: 58,000 + 4,000
During this next task, many students used a similar strategy as with 1,800 + 400. However, this time, they decomposed the 4,000 into 2,000 + 2,000: 58,000 + 2,000 + 2,000.
Task 3: 380,000 + 40,000
For the final task, one student Showed Multiple Ways of Jumping. Another student modeled how Starting with the Smaller Addend doesn't always mean more jumps. I also loved seeing student Making Connections between all three tasks (which supports Math Practice 8: Look for and express regularity in repeated reasoning).
Reasoning for Teaching Multiple Strategies
During this Addition and Subtraction Unit, I truly wanted to focus on Math Practice 2: Reason abstractly and quantitatively. I knew that if students learned multiple strategies of adding and subtracting numbers, I wouldn’t only be providing them with multiple pathways to learning, but I would also be encouraging students to engage in “quantitative reasoning” by “making sense of quantities and their relationships in problem situations.” By teaching students how to use a variety of strategies, such as using number lines, bar diagrams, decomposing, compensating, transformation, and subtracting from nines, I hoped students would begin to see numbers as units and quantities that can be computed with flexibility.
In order to provide students with guided practice, I created a PowerPoint presentation called, Compensation Practice Day 1. This way, I could intentinally provide students with a rigorous learning progression (instead of simply coming up with the numbers to compute during the lesson itself). I wanted students to start by adding and subtracting two-digit numbers (such as 34-29) and work up to six-digit numbers (such as 482,160 - 179,000). For today's lesson, we will focus on computing 2-digit to 4-digit numbers. During tomorrow's lesson, we will move on to 5-digit and 6-digit numbers.
Goal & Vocabulary
To begin, I showed students the first slide, which was the Goal: I can check the addition and subtraction algorithms using compensation. I explained: Sometimes we solve an addition or subtraction problem using the standard algorithm without actually knowing if we arrived at the correct solution or not. For this reason, it's important to use other strategies, such as compensation, to check the answer. To help students further understand the meaning of compensation, I introduced and explained the Compensation Poster. I also provided a real-life example of compensating: Compensation Meaning. Many students wanted to share their own experiences of begin compensated, such as having a meal compensated at a restaurant. Next, I showed students how to use compensation in math using the next slide: Compensation Example. Without providing students with an explanation of this slide, I simply asked: What do you notice? What do you think compensation is? The students quickly observed, "Compensation is when you add on to one addend to make it easier to work with."
We then moved on to the first problem, 34 + 29. I asked students: What could I add or subtract to make this an easier problem to solve? A student suggested, "Add one to the 29." I then Modeled how to use compensation to solve this problem while Students Showed Strategies on their white boards. Next, we moved on to 34 - 29. By using the same numbers, I was encouraging students to practice Math Practice 8: Look for and express regularity in repeated reasoning. Some students immediately pointed out, "I can still just add one... only this time I'm going to subtract." Many students compensated by adding one: 34 - 29 + 1 = 34 - 30 - 1. Others compensated adding 6: 34 +6 - 29 = 40 + 29 -6 (at first, mixing up addition and subtraction was a common mistake).
For independent practice time, I created 2 practice pages by copying & pasting portions of worksheets found at Math-Aids.com. I wanted to provide students with the space necessary to check the addition and subtraction algorithms using compensation: Compensation Practice Page 1. As students finished, they compared their answers with others at the back table.
During this time, I conferenced with as many students as possible. Often, I just looked over their shoulders to check for understanding: Student Practicing Independently. Other times, I would draw a number line model to help students understand when to add and when to subtract: Using a Number Line Model.