Transforming to Compute Larger Numbers
Lesson 14 of 16
Objective: SWBAT use transformation 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: 15,000 + 19,000
For the first task, many students started at 19,000 and took a jump of 1,000 to get to a landmark number, 20,000. Then, students took one more jump of 5,000: 15,000 + 19,000. While students shared their strategies with one another, I took this time to support a student: Providing Support With 15,000 + 19,000.
Task 2: 50,000 + 90,000
During the next task, some students took a jump of 50,000 to get to 100,000 (50,000 + 90,000) while others took a jump of 10,000 to get to 100,000. Either way, I was happy to see students using landmark numbers!
Task 3: 350,000 + 290,000
During this task, students used landmark numbers again. Some students started with the smaller addend while most other students began with the larger addend: 350,000 + 290,000.
Task 4: 500,000 + 900,000
For the final task, I asked students to explain the solution without representing their thinking on the number line. One student said, "1,400,000... because 900,000 + 100,000 equal one million. Then you addd 400,000 to get to 1,400,000."
Today's lesson is a continuation of yesterday's lesson, Transforming to Compute Smaller Numbers.
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 continue providing students guided practice using the transformation method, I created another PowerPoint presentation called, Transforming Part 2. This way, I could intentionally provide students with a rigorous learning progression (instead of simply coming up with the numbers to compute during the lesson itself). During yesterday's lesson, we focused on adding and subtracting 2-digit to 3-digit numbers using the transformation method. For today's lesson, we will move on to 4-digit to 6-digit numbers.
Goal & Review
At this point in the lesson, I asked students to join me on the front carpet with their number line boards (even though we would only be using this as a surface to write on instead of the number line itself). I wanted students to have a large work area. To begin, I reviewed the Goal on the first slide: I can check addition and subtraction algorithms using transformation. We also reviewed the meaning of transformation and examples of transformation by going back over slides from yesterday's lesson: Defining Transform, Defining Transformation, Transformation with Addition, and Transformation with Subtraction. We also took the time to review our conjectures about transforming from yesterday: Transformation with Addition and Transformation with Subtraction.
Rounding the Minuend or the Subtrahend
Next, I was hoping to inspire a conversation about rounding the minuend or the subtrahend. If you have 56 - 28, it is more purposeful to round the subtrahend: 56 - (28 + 2) because 30 is easy to take away from 56. Many students have been rounding the minuend instead: (56 + 4) - 28, which requires a harder calculation: 60 - 28. I wanted students to discover: taking away a multiple of ten is easier than subtracting from a multiple of ten.
So I introduced the following slide to students: Rounding the Minued or Subtrahend. I explained: I am wondering... Which is better to round when you are using the transformation method... the minuend or the subtrahend? Pointing to the slide, I said: Here's an example of rounding the minuend. Here's an example of rounding the subtrahend. I'm going to give you some time to just observe. I want you to construct an argument in support of rounding to the minuend or the subtrahend. After a few minutes, hands began to shoot up in the air, many times with a, "Oh! I know!" Providing students with opportunities to construct evidence-based arguments is an important part of developing Math Practice 3: Construct viable arguments and critique the reasoning of others.
A great conversation followed. Each time a student shared their thinking, such as, "Rounding the subtrahend makes the number easier to subtract," I would ask: What is your evidence? How do you know. Then, the student would say, "Because 58 - 40 is easier that 60 - 42." By the end of the discussion, I asked: Who thinks you should round the minuend when transforming? (No hands, if any went up.) Who thinks you should round the subtrahend when transforming? (Almost all hands went up.)
Transforming When Adding
I explained: Yesterday, we used the transformation method to solve addition and subtraction problems involving smaller numbers. Today, we are going to use transformation to solve addition and subtraction problems involving larger numbers!
First, let's take another look at the addition conjecture we came up with yesterday: If you take away a number from one addend, you have to give it to the other addend. We also agreed that the reverse would also be true: If you add a number to one addend you have to subtract it from the other addend. Let's continue testing this conjecture today with larger numbers to see if it really is true! (Throughout the next activity, I'll keep referring to this conjecture and asking students: Do you think our conjecture is still true? How do you know?)
I began by showing the first problem: 3,159 + 1,402. I wanted to review the steps of transformation so I Modeled 3,159 + =1,402 on the Board. I asked: How could we use the transformation strategy to make this problem easier to solve? One student suggested, "Take away 402 from 1,402." So I subtracted 402 from 1,402 and I added 402 to 3,159 by decomposing the 402 into a 400 and 2. We then added 3,561 and 1,000 to get 4,561. Some students were still grappling with the idea that we take away from one added and give it to the other so I modeled a simpler problem: 10 + 9. I then added one to the 9 and gave it to the 10, which results in 9 + 10. Many students said, "Oh... now I get it." I then demonstrated the following a horizontal number sentence: 10 - 1 + 9 + 1 and asked: What is a negative one plus a positive one? If I have one cookie and I eat one cookie, how many cookies will I have left? Students giggled and said, "Zero!!!" Then I crossed off the -1 and the 1 in the equation to get back to 10 + 9. Even more students said, "Oh.... yeah!"
I then gave students more time to come up with their own strategies. During this time, some students found Multiple Strategies to Solve 3,159 + 402. Here, a student explains the strategy that I modeled on the board: Student Explains How to Transform 3,159 + 1,402. Following each opportunity for students to come up with more strategies, I asked students to turn and talk with a partner.
Following the same process (teacher modeling + students solving on boards + discussion), we then moved on to the next two addition problems: 16,003 + 12,095 and later on: 132,999 + 126.057. I loved the range of strategies that students used. Here, one student Adjusting Each Addend by 95 whereas another student Adusting each Addend by 6,003. It was so powerful for students to arrive at the solution in a variety of ways.
Transforming When Subtracting
Before moving on to subtraction, we reviewed our Conjecture about Subtraction from yesterday: If you subtract a number from the minuend, you must subtract that same number from the subtrahend. We also agreed the following was also true: If you add a number to the minuend, you must add that same number to the subtrahend. Again, I said: Let's continue testing this conjecture today with larger numbers to see if it really is true! (Throughout the next activity, I'll keep referring to this conjecture and asking students: Do you think our conjecture is still true? How do you know?)
We then completed three subtraction problems together as a class: 3,159 - 1,402, 16,003 - 12,095, and 132,999 - 126,057. Following the same procedures as before, I modeled the transformation process with each problem. Then students used the transformation process to solve each problem using other strategies. This would be followed by student discussions.
Here, a student explains how to use Transforming to Subtract 132,999-126,057. Many students resorted to adjustments of 1-digit numbers (such as adding or subtracting 1). It took time and encouragement for students to try other strategies. After discussing this problem further, the student decided to try Rounding the Subtrahend Instead.
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 transformation: Transformation Practice.
To get students started and to provide clear expectations, I modeled the first problem, 1371 + 3586, while students solve the problem on their papers.
As students finished, they compared their answers with others at the back table or within their group.
During this student practice time, I conferenced with as many students as possible. Often I would ask students:
- Is there another way you could have transformed this problem?
- How is this strategy helpful to you as a mathematician?
- Is it better to round the subtrahend or the minuend?
Here's a Student Example Page 1 and Student Example Page 2. Most students understood and were able to apply the transformation strategy. However, I would like to see some students apply this strategy more effectively. For the problem 7612 + 4986, instead of just subtracting 2 from 7612, I'd like to see students subtract 12 or even 612. We will continue to practice this strategy throughout the year so that students can further hone their skills.