Using Mental Math to Divide

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Objective

SWBAT use basic facts to solve division problems with 3-digit dividends and 1-digit divisors.

Big Idea

Using basic facts can help you divide 3-digit dividends and 1-digit divisors.

Opener

5 minutes

In today's lesson, the students learn to use basic facts and patterns with zeros to solve division problems.  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.

To get started, I ask the students a question.  "Who remembers the relationship between multiplication and division?"  I give the students a few minutes to think about the question.   I take a few student responses. One student replies, "Multiplication is related to addition, and division is related to subtraction."  The second student replies, "Multiplication makes a number go up, and division makes a number go down."  I tell the students that they are both right, but I am looking for another word.  Finally one student responds, "Multiplication and division are inverse of each other."  Now, to make sure they understand the word "inverse", I ask what 'inverse' means.  A student finally tells me that they "undo" each other.  I let the students know that today, we will use multiplication and arrays to help us solve division problems. I tell the students that this skill is similar to the skill they learned earlier when they multiplied by multiples of 10 and 100.  

Whole Class Discussion

10 minutes

I call the students to the carpet as we prepare for a whole class discussion.  The  Using Mental Math to Divide 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.  

We begin with a sample division problem.  

14 ÷ 2 = 7

140 ÷ 2 = 70

1,400 ÷ 2 = 700

I tell the students to look for patterns.  We can use the basic facts to help us divide.  If your divisor is 2, then use the 2’s to help.  2 x ? = 14.  2 x 7 = 14.  So, 7 is the quotient.  If you are dividing multiples of 10 or 100, then use the basic facts, then count the zeros.  

Let's try another example:

48 ÷ 6 =  8

Use your basic facts for your 6’s.  6 x ? = 48.  6 x 8 = 48.  From there, use a pattern to help you with multiples of 10 and 100.

480 ÷ 6 =  ?

If 48 ÷ 6 = 8, what is 480 ÷ 6? 

The students say that the product is 80.  I call on one student to explain to me how he knows that the product is 80.  The student responds that because 48 ÷ 6 = 8, all I have to do is add 1 zero.

I let the student know that he is correct.  I go on to explain to the students that this is a short cut method.  Because 80 x 6 = 480, we know that 480 ÷  6 = 80.

4,800 ÷ 6 =  ?

If 48 ÷ 6 = 8, what is 4,800 ÷ 6? 

I call on a student to give me the answer.  The student tells me that the answer is 800 because 48 divided by 6 = 8, and then you add 2 zeros.

To show the connection between multiplication and division, I answer the question, "How does multiplication help with division?"

If you know your multiplication facts, you can easily find the quotient for the basic facts.  We can use an array to help us.

48 ÷ 6 = 8

x x x x x x x x

x x x x x x x x

x x x x x x x x

x x x x x x x x

x x x x x x x x

x x x x x x x x

It is easy to find the quotient for the multiples of 10 and 100.  Once you divide the basic facts, then count the zeros to multiply by multiples of 10 and 100.

Another way to model:

You can also make “groups” in order to show division.

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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.  The students must work together to find the quotient to division problems. They must look for a pattern (MP7).  They must make a model of the problems (MP4).  They must communicate precisely to others within their groups. They must use clear definitions and terminology as they precisely discuss this problem

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.  How does your model represent your problem?

2. What pattern do you notice?

3.  How did you use multiplication to helpyou with the division problem?

As I walked around the classroom, I heard the students communicate with each other about the assignment.  From the video, you can hear the 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. 

Based upon the sample Student Work, it shows that the students were able to divide 56 by 7 to come up with 8 as the quotient.  From that answer, the students knew that 560 divided by 7 = 80, and 5,600 divided by 7 = 800.  This is evident that the students understood the skill.

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.funbrain.com/cgi-bin/mb.cgi?A1=start4&A2=1&ALG=No&Submit=Play+Ball

Closure

15 minutes

To close the lesson, I have one or two 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 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.  

Student Observation:

Most of the students really got this skill.  I feel that it is because we used multiplication charts to help us.  But, this skill is similar to the skill when they multiplied by multiples of 10 and 100.  The students took what they learned in that skill and used it for this skill.