Estimating Quotients

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Objective

SWBAT use compatible numbers and rounding to estimate quotients.

Big Idea

You can substitute numbers that can help make estimating division problems easier.

Opener

5 minutes

In today's lesson, the students learn to use compatible numbers and rounding to estimate quotients.  This aligns with 4.NBT.B6 because the students find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. It is important for the students to know how to estimate because sometimes they may not need to find an exact answer for a problem.  Also, estimation is an excellent way for the students to check their quotients to see if it is reasonable.  

To get started, I ask the students a question.  "How do you estimate numbers?"  I give the students a few minutes to think about the question.   I take a few student responses. One student says, "By rounding."  I let the students know, "Today, we will use compatible numbers  to estimate quotients.  Our multiplication facts will help us."

Whole Class Discussion

10 minutes

I call the students to the carpet as we prepare for a whole class discussion.  The ESTIMATING QUOTIENTS power point is already up on the Smart board.  I like for my students to be near so that I can have their full attention while I'm at the Smart board.  

I begin by going over important vocabulary for this lesson.  The students will have to know these terms to understand the lesson.

Vocabulary:

compatible numbers -  numbers that are close in value to the actual numbers, and which make it easy to do mental arithmetic.

estimate - an approximate calculation

Problem 1:

Terri has 412 pieces of candy.  She wants to share the candy equally with 7 friends.  If her 7 friends get the same amount of candy, about how many pieces of candy will they each receive?

Let me model this problem for you.

First, I ask the students to identify any clue words that will help them solve this problem. One student lets the class know that "share the candy equally" tells us to divide. I let the student know that he is correct.  I go on to explain that based upon past knowledge, the clue words "share equally" is a clue to divide.  Therefore, this is a division problem. The problem is 412 divided by 7. Also, the key word "about" let us know that we are going to estimate.  

Because we do not need an actual amount, we can use our multiplication chart to help us estimate our number.  We are dividing the candy among 7 friends.  Each friend will get the same amount of candy.  A strategy that can be used is to is to find multiples of 10 that can be easily divided by 7.   Let's look at the multiplication chart.  (I have the multiplication chart pulled up on my Smart board.)  We know that a multiple of 10 ends with 0 in the ones place.  Because we are working with a 3-digit number, we know the last digit will be 0 when we round the number.  Therefore, underline your first 2 numbers in the dividend.  In this problem, the first two numbers are 41.  Look at the multiplication chart to see 7 x ___ equals a number that comes close to 41.  I see that 7 x 6 = 42.

420 is easily divided by 7 because 7 x 6 = 42, therefore 420 divided by 7 = 60.

Let's try another one.  

Problem 2:  268 divided by 3.

Let's look at the multiplication chart.  Again, we know that a multiple of 10 ends with 0 in the ones place.  Because we are working with a 3-digit number, we know the last digit will be 0 when we round the number.  Therefore, underline your first 2 numbers in the dividend.  In this problem, the first two numbers are 26.  Look at the multiplication chart to see 3 x ___ equals a number that comes close to 26.  I see that 3 x 8 = 24 and 3 x 9 = 27. 

Which multiplication problem can we use for the compatible number? 

I let the students yell out to tell me which number they think.  Some students yell 24 and the others yell 27.

I let them know that because both numbers are close to 26, we can use either problem for the compatible number.  Remember, a compatible number is a number that is close in value to the actual number.

We can estimate 268 = 240 or 268 = 270.

Group or Partner Activity

20 minutes

I give the students practice on this skill by letting them work together.  I find that collaborative learning is vital to the success of students.  Students learn from each other by justifying their answers and critiquing the reasoning of others.

For this activity, I put the students in pairs.  I give each group a Group Activity Sheet Estimating Quotients.  The students must work together to estimate the quotients using rounding or compatible numbers.  Students that are not familiar with their multiplication facts were able to use a multiplication chart to help with this skill (MP5).  The Multiplication Chart is very important in this activity to help the students find a compatible number to use to estimate the quotient.  They must communicate precisely to others within their groups. They must use clear definitions and terminology as they precisely discuss this problem (MP1).

The students are guided to the conceptual understanding through questioning by their classmates, as well as by me.  The students communicate with each other and must agree upon the answer to the problem.  Because the students must agree upon the answer, this will take discussion, critiquing, and justifying of answers by both students.  From the video, you can hear the students discuss the problem and agree upon the answer to the problem.  As the pairs discuss the problem, they must be precise in their communication within their groups using the appropriate math terminology for this skill.  As I walk around, I am listening for the students to use "talk" that will lead to the answer. I am holding the students accountable for their own learning.  

As they work, I monitor and assess their progression of understanding through questioning. 

1.  What is the dividend in this problem?

2. What multiplication problem will help estimate the dividend?

3.  Is that the only estimated quotient for this problem?  Why or why not?

As I walked around the classroom, I heard the students communicate with each other about the assignment.  There is classroom chatter and constant discussion among the students.  Before Common Core, I thought that a quiet class working out of the book was the ideal class.  Now, I am amazed at some of the conversation going on in the classroom between the students. 

 Any groups that finish the assignment early, can go to the computer to practice the skill at the following site until we are ready for the whole group sharing: http://www.aaamath.com/est32_x2.htm

 

My Observations:

Using compatible numbers is usually a skill I do not look forward to teaching.  It is a hard concept for the students.  In this lesson, it was no exception.  The students found it difficult to find the compatible number.  They were comfortable with rounding, so they preferred to round the number.  However, that did not work in some division problems.  Therefore, I required the students to use the multiplication chart and try.  To keep the students from being discouraged, I explained to them that this is the first time that they are using this skill.  I remind them that the more you practice a skill, the easier the skill gets.

 I gave the students their 20 minutes to work on this activity, as I walked around and helped the pairs by using questioning.  By the end of the 20 minutes, a few of the pairs actually could find the compatible numbers.  

 

Closure

15 minutes

To close the lesson, I have students share their answers.  This gives those students who still do not understand another opportunity to learn it.  I like to use my document camera to show the students' work during this time.  Some students do not understand what is being said, but understand clearly when the work is put up for them to see.

I feel that by closing each of my lessons by having students share their work is very important to the success of the lesson.  Students need to see good Student Work samples, as well as work that may have incorrect information.  More than one student may have had the same misconception.  During the closing of the lesson, all misconceptions that were spotted during the group activity will be addressed whole class.  

 

Misconceptions:

Some students wanted to use rounding because they were comfortable with rounding.  But, they found that the divisor could not easily go into some of the numbers that they rounded.  By the end of the lesson, the students understood how to use the multiplication chart to find a multiplication fact that could be used to help them find the compatible number.