Transforming to Compute Smaller Numbers
Lesson 13 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.pdf. For this Number Talk, I am encouraging students to represent their thinking using an open number line model.
Task 1: 50 + 90
For the first task, most students started with the larger addend, 90, and took one jump of 10 to get to a landmark number, 100. Then, they took a jump of 40. This student took two jumps of 20 after reaching 100: 50 + 90.
Task 2: 500 + 900
For the next task, most students took a jump of 100 to get to a landmark number, 1000. Here, a student then takes two jumps of 200 to get to the sum: 500 + 900.
Task 3: 2,500 + 2,900
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 using the transformation method, I created a PowerPoint presentation called, Transforming Part 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 2-digit numbers (such as 78 + 14) and work up to six-digit numbers (such as 132,999 - 126,057). For today's lesson, we will focus on computing 2-digit to 3-digit numbers. During tomorrow's lesson, we will move on to 4-digit to 6-digit numbers.
Goal & Vocabulary
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 showed students the first slide, which was the Goal: I can check addition and subtraction algorithms using transformation. I explained: Just as we have talked about in past lesson, it's important to use multiple strategies, such as the standard algorithm, compensation, or transformation, to make sure our solutions are accurate. To help students further understand the meaning of transformation, we first discussed the meaning of transform: Defining Transform. Once students understood the meaning of transform, we moved on to Defining Transformation. Without providing students with an explanation of this slide, I simply asked: What do you notice? What do you think transformation is? I waited until almost all hands were excitedly raised to call on a student. One student said, "It's when you take an amount from one number (addend) and and give it to the other number (addend)."
Making a Conjecture about Addition
We then moved on to further Analyzing Transformation with Addition. Using this slide, I showed examples of transformation with addition and asked students to help make a conjecture about addition. I reminded them: Remember, a conjecture is a judgment based on inconclusive or incomplete evidence. For example, I might say that Riley likes the color green because he is wearing green today, but I need to gather more evidence to truly prove this to be true. Turn and talk: What is a conjecture you can make about transforming addition? After giving students time to discuss their observations with partners, we then discussed conjectures as a whole group and came up with the following Conjecture about Addition: 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.
Testing the Conjecture
Knowing that making conjectures and exploring the truth of conjectures is an important component to Math Practice 3 (Construct viable arguments and critique the reasoning of others), I then explained to students: Now we are going to test this conjecture. Let's see if it always works and if it really a true judgement! I showed students the first problem, 36 + 21 and asked: What could we add or subtract from one added to make this an easier problem to solve? A student suggested, "Subtract 1 from 21." I then asked: According to our conjecture, if we subtract one from 21, what else do we have to do? Students responded, "Add one to 36!" Okay! Let's test our conjecture and see if it really works! Here, I am Modeling 36 + 21 while Students Modeled 36 + 21. on their number lines. After we finished I asked: So far, is our conjecture true? Students were quick to say, "YES!"
Next, we moved on to 78 + 14.. We continued the same process as above. Students "Tested" the Conjecture on their number lines and, as a class, we discussed how to use transformation to solve this problem: Modeling 78 + 14..
We then solved, discussed, and modeled each of the remaining addition problems (99 + 144. and 609 + 247.). Many times students divided their boards up so they could show multiple ways of transforming each addition problem. Here a student shows her first Two Strategies and Two More Strategies! While some strategies were more efficient than others, it was important for students to experiment in order to discover which strategies were the most helpful.
Following the last addition slide (609 + 247), I then asked students: Have we proven our conjecture to be true or not true? Students responded, "It's true!" How do you know? (Because we proved it to be true by testing it!)
Making a Conjecture about Subtraction
Students were ready for Analyzing Transformation with Subtraction. Using this slide, I again showed examples of transformation, only this time, with subtraction. I asked students to help make a conjecture about subtraction. Turn and talk: What is a conjecture you can make about transforming addition? After giving students time to discuss their observations with partners, we then discussed conjectures as a whole group and came up with the following Conjecture about Subtraction.: 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.
Testing the Conjecture
Students were ready to test their conjectures! I showed students the first problem, 34 - 19 and asked: What could we add or subtract from the subtrahend or minuend to make this an easier problem to solve? A student suggested, "Add 1 to the 19." I then asked: According to our conjecture, if we add one to 19, what else do we have to do? Students responded, "Add 1 to 34!" Okay! Let's test our conjecture and see if it really works!
Following the same process as before, students solved 34 - 19 on their own boards while I Modeled 34 - 19 on the Board.
We then solved, discussed, and modeled each of the remaining addition problems (61 - 38.jpg, 137 -112, and 609 - 247) to see if our conjecture was true! Just as before, students experimented with Multiple Strategies. I knew that this activity was not only helping students practice transformation, but also helping build number sense.
Following the last addition slide (609 - 247.), I then asked students: Have we proven our conjecture to be true or not true? Students responded, "It's true!" How do you know? (Because we proved it to be true by testing it!)
Now that students discovered the meaning of transformation by investigating examples, I felt comfortable introducing the Transformation Poster to help students during student practice time.
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 A.
To get students started and to provide clear expectations, I modeled the first problem, 59 + 76, while students solved 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?
- Can you explain why you took away 5 from both the minuend and the subtrahend?