Finding Compatible Numbers to Check Subtraction

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Objective

SWBAT use compatible numbers to check for reasonableness when subtracting multi-digit numbers.

Big Idea

Students will round the subtrahend and minuend to numbers that work well together to determine if an answer is reasonable.

Opening

15 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 use their knowledge of decomposing to subtract.

Tasks 1: 89 - 56

For the first task, students were able to successfully decompose to solve: 89-56 Student Example A. This student surprisingly checked for reasonableness: 89-56 Student Example B. However, here's the student that surprised me the most: 89-56 Student Example C. I loved how he came up with: 89-56 = (40-20) + (40-30) + (5-3) + (4-3). Others were so inspired that they immediately tried using his strategy!  

Tasks 2: 689 - 56

During the next task, students decomposed in a variety of ways: 689-356 Student Example A. I really enjoyed watching this student replicate another student's strategy: 689-356 Student Example B

 

 

 

Teacher Demonstration

40 minutes

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. 

Goal

I began today's lesson by explaining the goal written on the board: I can use compatible numbers to check for reasonableness when subtracting multi-digit numbers. I explained: Compatible numbers are numbers that work well together. I wrote an example problem on the board: 21 +28 on the board and continued: For example, if I had the problem 21 + 28, I could think of the 21 as being close to 25 and the 28 as close to 25 too. If I add 25 + 25, I'll know that a reasonable answer should be around 50. Now, what is the exact solution to 21 + 28? Students responded quickly, "49!" Is 49 close to 50? "Yes!" Then this tells me that my answer is... Students piped in, "Reasonable!" 

Vocabulary

We then moved on to vocabulary development. I first began by reviewing key vocabulary from yesterday's lesson (addend, sum, algorithm, and checking for reasonableness). Then, I moved on the today's key vocabulary: Today, I want to continue using high-level math vocabulary when you are turning and talking about the reasonableness of answers. By teaching math vocabulary, students will have the tools to truly practice MP 3 (Constructing Viable Arguments). 

I then taught students the terms, difference, minuend, and subtrahend using the Subtraction Vocabulary Poster. After teaching new vocabulary, I always give students time to absorb the new information by turning and talking: Tell someone next to you what the minuend is! 

Finding Compatible Numbers

I explained: Remember, today we are going to be finding compatible numbers (numbers that work well together) to check for reasonableness. Let's practice this together! Using the  Checking for Reasonableness Poster, we discussed how to make sure answers are not too high and not too low.

Guided Practice

I continued: As I model each problem on the board, please use your white boards to practice finding compatible numbers with me! Prior to the lesson, I created a list of problems (with increasing complexity) on the board prior to the lesson:

  • 92-52
  • 112-48
  • 156-49
  • 124-26
  • 572-53
  • 1291-48

Pointing to the first problem, I asked students: What compatible numbers could we use to check the solution to this problem? Hands shot up quickly, almost as if it was a game! One student said, "92 is close to 100." Another offered, "52 is close to 50." I wrote the the "approximately equal to sign" (below) followed by 100 - 50. Okay, everyone, what it is 100 - 50? "50!" Is this the exact answer? "No" What is the exact answer? "40!" Is 40 close to 50? There were mixed views on this question. Some students felt that 40 and 50 were not close at all! So then I asked, Well, what if we had gotten 124 as a solution. Would we know that our answer is reasonable or unreasonable? We continued in this same fashion, solving each problem on the list: Compatible Numbers Practice.

 

Then, I asked: Do you think you are ready to practice this on your own? The room filled with excitement! 

Student Practice

45 minutes

To help students meet today's goal, I wanted them to practice solving the algorithm and checking their answers by finding compatible numbers so I choose the following practice page from the Grade 4 Module 1 Engage New York Unit found online: 

Practice Page: Subtraction Algorithm Practice

Algorithm Practice

At first, I asked students to solve each problem using the standard algorithm. To make sure all students remembered the steps of subtraction, we solved the first row altogether, step-by-step. Then, I asked students to continue on with group members. 

During this time, I conferenced with students and checked for understanding. Some students needed extra support with borrowing across zero, remembering to subtract instead of add, and/or subtracting down instead of subtracting up.

Once finished, I asked students to check their work with a partner. I do this specifically to support Math Practice 3 (Construct Viable Arguments) and to provide students with opportunities to discuss possible mistakes when they have arrived at two different answers.

Practice Finding Compatible Numbers to Check for Reasonableness 

Next, I modeled (Teacher Model) how to Use Compatible Numbers to check each problem in the first row for reasonableness. Thereafter, students got right to work!

During work time, I asked guiding questions, such as: How do you know your answer is reasonable? Here's an example of a student conference during this time: Finding Compatible Numbers.

The most challenging part of completing this page was the last problem. Since I hadn't taught my students how to use a tape diagram (bar diagram) yet, I chose to model this on the board: Teacher Support with Last Problem. Then, students were able to find success: Student Example of Last Problem.

Again, as students finished checking each problem for reasonableness, they checked their answers with a partner. Most students were able to reach the goal: Student Practice Page Example