Estimating Quotients Using Compatible Numbers

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Objective

SWBAT estimate quotients of two-digit divisors and 2 -4 digit dividends using compatible numbers.

Big Idea

To estimate quotients with 2-digit divisors, we may need to change both the dividend and the divisor to be compatible numbers. We want to think of numbers that are close to the original numbers and use basic division facts we already know.

Think About It

7 minutes

Students work independently on the Think About It problem.

After 3 minutes of work time, I ask students what they needed to do with this problem.  I first want students to identify that the word 'about' signals estimation.   Many students will have rounded the numbers in this problem to 160 divided by 30.  I  use this for the basis of our conversation.

I'll ask students why we estimate, and why rounding as an estimation strategy didn't make mental math easy for us in this problem.  Students will talk about how 30 can't be divided into 160 evenly.

I'll ask if anyone decided to use 150 divided by 30 as an estimate, and have that student justify the choice.  If no one did this, I'd ask 'what if we used 150 divided by 30 as our estimate' to get the conversation started.

I'll frame the lesson by telling students  that in this lesson we are going to estimate quotients of 3 and 4 digit numbers by divided 2-digit numbers so that we can estimate a problem like the one I just gave you in our heads.  We’ll use compatible numbers to do this. In our next lesson, we will find exact answer and this skill will help us with that as well.

Intro to New Material

10 minutes

To start the Intro to New Material section, I have a student read the first problem aloud and guide the class through the needed annotations.  At this point in the year, in my classroom, the annotations are relatively simple:  students box important numerical information, underline the question, and add any needed additional annotations for sense-making.  In this problem, for example, they'd write ≈ above the word 'about.'

I ask students what would happen if we rounded each number to the nearest 10 in this problem.  I have students vote with their thumbs about whether or not we should use 430 ÷ 60. I call on a student to share out justification for not using expression.

I guide students through following these steps to use compatible numbers:

1)      Round to the nearest 10.  If it is a basic fact, divide.  If not:

2)      Underline the first two digits in the dividend and the first digit in the divisor.

3)      Draw an arrow down from the divisor and re-write the digit.

4)      Look at the first two digits in the dividend.  Mentally decide which is the closest multiple of the first digit in the divisor. 

5)      Rewrite the dividend with the basic fact and include a 0 in the ones place (and tens place if it is a 4-digit number)

6)      Divide mentally using powers of 10.

7)      Check the reasonableness of your estimate, using multiplication.

 

I have students guide me through the annotations and estimation for problem #2, and I focus on the organization of the work space.  I will insist that students include annotation, the actual expression, the compatible numbers expression, and a multiplication check.

I've included three division problems without any real-world context in this section.  I'll use those if I think the class needs more practice with me before moving on to partner practice.  If I sense the class does not need this support, we'll skip them.

Partner Practice

15 minutes

Students work in pairs on the Partner Practice problem set.  As students work, I am circulating around the room and checking that all work includes annotation, the actual expression, the compatible numbers expression, and a multiplication check.  I'm also looking for:

  • Are students using strategies for dividing by multiples of 10?
  • Are students explaining their thinking to/with their partners?
  • Are students using compatible numbers to divide?

 

I'm asking:

  • Explain you determined this answer. 
  • How did you use a basic fact to help you solve?
  • How did you check your work? 
  • Why didn't you...  (ask about an expression that includes legitimate rounding, but does not provide an expression that's easy to calculate)

 

After 10 minutes of partner work time, I ask students to vote on which teacher is correct in Problem 4.  I then ask 1-2 students to read their written responses to this question.  Students complete the Check for Understanding problem independently, and I circulate to check work.


Independent Practice

15 minutes

Students work on the Independent Practice problem set. 

The set is designed to get increasingly more rigorous as students progress through the problems.  Problems 1-4 are fairly straightforward. In Problem 5, a constraint from the previous problem is changed, and students have to decide whether or not to adjust their estimates.  Problem 6 requires students to decide whether someone's estimate is reasonable.  Students must come up with their own estimates, and then write to disprove Brenna.  In Problem 7, if students are not careful readers, they may work with 294÷3, rather than 294÷21.

Closing and Exit Ticket

8 minutes

After independent work time, I display student work on the document camera that uses 294÷3 for Problem 7.  I teach this lesson during the 2nd week of the school year, so I use this as an opportunity to reinforce that we all make mistakes in math, and that we can all learn from one another's mistakes (I also have my own created work sample, in case no one has made this mistake).  I'll ask students to identify the mistake and suggest what the corrected work should look like.

Students then complete the Exit Ticket independently to close the lesson.  I've included a sample of what student work could look like.