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 continue trying the "subtract from nines" strategy as well as other chosen strategies.
Task 1: 100 - 13
For the first task, students solved 100 - 13. I encouraged students to at first try the "subtract from nines" strategy from yesterday's lesson. I knew that extra practice would help students become more proficient with this strategy. Some students went on to use a variety of other strategies: 100-13.
Task 2: 100 - 43
During the next task, 100 - 43, most students turned the 100 into 99 + 1 and then subtracted 99 - 43 to get 56. Then they added the one back in to get a final answer of 57.
Task 3: 1000 - 273
For this task, students used various strategies. For example, this student, solved 1000 - 273 using the subtract from nines method, an open number line, and decomposing. Here, a student explains how she solved this problem by decomposing, using landmark numbers, and Adding up from 273 to 1000 in order to subtract.
Task 4: 10,000 - 9,456
For the final and most complex task, 10,000 - 9,456, students changed the 10,000 into 9,999 + 1 and then subtracted 9,999 - 9,456. It was great to see students subtracting from the left to right, knowing that borrowing wouldn't be necessary. Finally, students added the one back in to get to 544.
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.
Prior to beginning the lesson, I invited students to join me on the front carpet with white boards and markers.
Goal & Strategy Poster
I began today's lesson by explaining the goal. Fourth graders, today's math goal is very important: I verify the accuracy of algorithms. I asked students: Does anyone know what verify means? After a few responses, I clarified: To verify means to check for correctness. For example, when Aaron and I are traveling, I'll often call ahead of time and verify that our hotel room is still book and that we have the ocean view that we requested!
I then presented the Verifying the Algorithm Poster and continued: In math, verifying is the process of making sure your solution is accurate. We continued by discussing the importance of verifying the accuracy of algorithms. By teaching students how to verify their answers, I was supporting the development of Math Practice 6: Attend to Precision, as "calculating accurately and efficiently" is a key component to preciseness.
Modeling & Practicing Verification with Addition
Next, I modeled 10 + 3 = 13. I explained: If 10 + 3 = 13, then I can verify my solution of 13 by taking the sum and subtracting 10 to get 3, the other addend. Students quickly pointed out, "You can subtract the 3 to get 10 too. " Modeling this on the board, I asked, So which addend should you always subtract... the first one or the second one...? Students responded, "It doesn't matter." I knew this was a really important point so I asked students to turn and talk to explain why we could subtract either addend when verifying addition. I was pleased to hear many students referring to the "evidence" on the board to construct viable arguments (Math Practice 3).
We moved on to a larger problem: 589 + 134. We labeled key vocabulary to begin with (sum and addends). Then, as I solved the problem using the standard algorithm on the board, students also completed the problem on their white boards. After arriving at a solution, I asked, What should we do to verify the solution? Students reflected upon the previous problem and said, "Subtract either addend from 723!" Altogether, we verified the algorithm by subtracting 723-134. Just to prove the point, we also subtracted 723-134.
I wrote on the board, "To Verify Addition" and placed a box around this title. I asked, What rule do we always follow when verifying addition? "We take the sum and subtract either addend!"
We then practiced a few more problems together and applied our new rule: 59 + 26, 859 + 265, and 8592 + 2655. This time, I asked students to try solving each problem on their own first. As students finished, I encouraged turning and talking about their work (which supports Math Practice 3). This also gave me an opportunity to work conference with and support students.
Modeling & Practicing Verification with Subtraction
At this point, students were ready to learn about verifying subtraction. I began by modeling 10 - 3 = 7. I explained: If 10 - 3 = 7, then I can verify my solution of 7 by taking 7 and subtracting 3. (I decided to provide students with non-examples first. Students immediately objected, "No! That equals 4! That doesn't help! You need to add!" I continued by giving another non-example: Okay.... okay... then you take 7 and add 10. "No! That equals 17! You need to add 7 + 3 to get 10!" I then modeled 7 + 3 = 10 and explained: Oh... now I get it! You take 7 and add 3 to get 10. What is the 7 called again? ("The difference!") And what is the 3 called again? (Students referred to our math vocabulary wall, "The subtrahend!") Then what is the 10 called? ("The minuend!") So I take the difference and add the subtrahend to get to the... Students continued my sentence, "Minuend!"
We moved on to a larger problem: 95 - 62. We labeled key vocabulary to begin with (minuend, subtrahend, and difference). Then, as I solved the problem using the standard algorithm on the board, students also completed the problem on their white boards. After arriving at a solution, I asked, What should we do to verify our solution? Students reflected upon the previous problem and said, "Add the difference to the subtrahend!" Altogether, we verified the algorithm by adding 33 + 62. Just to prove the point, we also added 33 + 95. Students decided that you must always add the subtrahend to the difference and never the minuend!
I wrote on the board, "To Verify Subtraction" and placed a box around this title. I asked, What rule do we always follow when verifying subtraction? "We take the difference and add the the subtrahend!" I pointed out a few differences between our rules to encourage students to make connections: So when we verify addition we use which operation? "Subtraction!" And when we verify subtraction, we use which operation? "Addition." And when verifying an addition problem, we can subtract either.... (Students filled in the blank, "Addend!") But when verifying a subtraction problem, we always add the... ("Subtrahend!")
We then practiced a few more problems together and applied our new rule: 951 - 628 and 9515 - 6288. This time, I asked students to try solving each problem on their own first. As students finished, I encouraged turning and talking about their work (which supports Math Practice 3). Again, this gave me an opportunity to work conference with and support students.
To provide students with some guided practice before verifying addition and subtraction problems independently, we played Multi-Digit Number Battle as a whole class (boys verses girls). I found this game in a card game resource: Playing Cards Math.
Adding Multi-Digit Numbers
At first, we began by playing an addition version of this game. Each team was given two sets of two cards to create two 2-digit addends. One student from each team rearranged the cards to create two 2-digit addends. Then, All Students Solved the Problem using the addition algorithm and the verification strategy to find the sum of their cards. Whichever team had the greatest sum got to take all the cards.
Subtracting Multi-Digit Numbers
Later, we moved on to subtracting 2-digit and 3-digit numbers to get the greatest difference using the same procedures. This little game was a fun and refreshing way of practicing the verification strategy! Students wanted to solve each problem accurately for the "sake of winning" the game.
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 verify the addition and subtraction algorithms: Practice Verifying.
To get students started and to provide clear expecatioans, I modeled the first problem while students solve the problem on their papers.
During this student practice time, I Conferenced with as many students as possible. Often I would ask students:
As students finished, they compared their answers with others at the back table or within their group.