##
* *Reflection: Developing a Conceptual Understanding
Estimating Products - Section 4: Closure

Using a number line is a great way to give the students a conceptual understanding of rounding. In the past, I used the rules of rounding. I found that the students had no idea what they were doing. They only followed the steps, but did not know why. By using the number line, the students understand that they are rounding the number to a place value, and they know the 2 numbers that they must consider. The number line gives the students a visual of what rounding actually means.

*Using Number Lines*

*Developing a Conceptual Understanding: Using Number Lines*

# Estimating Products

Lesson 11 of 23

## Objective: SWBAT use rounding to estimate the product in multiplication problems.

*55 minutes*

#### Opener

*5 min*

In today's lesson, the students learn to estimate when they do not need the exact answers and to check to see if their answers are reasonable. This aligns with **4.OA.A3** because the students assess the reasonableness of answers using mental computation and estimation strategies including rounding.

To relate this lesson to their every day lives, I ask the students a question. "Is there ever a time that you may need to estimate?" I call on a few students to share their ideas about when they would need to estimate. The students came up with ideas such as guessing at the number of jellybeans in a jar or the number of students on a bus. By providing this information, the students can see how this lesson is important in their lives.

"Today, you learn to estimate greater numbers. In previous lessons we have estimated smaller numbers, but you need to know how to estimate larger numbers as well. What are some things that you may need to estimate that are large numbers?" I take a few responses from the students. Some student responses: people or cars on the street. I let the students know that it is important to understand the concept of the lesson. Therefore, we will use number lines to help us estimate. I go on to tell the students that a number line is an excellent tool to use because it gives you a visual picture of which number is closest to the number you are trying to round.

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#### Whole Class Discussion

*15 min*

I like for my students to come to the carpet when I am teaching from my Smart board. This allows every student the ability to hear and see what is being said and done on the Smart board.

I start the lesson by discussing what the students already know about rounding. The students are all familiar with the rules of rounding. Underline the place to be rounded. If the place behind the underlined digit is 5 or more, the underlined digit goes up by 1 and everything behind it become zeros. If the place behind the underlined digit is 4 or less, the underlined digit stays the same and everything behind it become zeros.

However, for this lesson, I want the students to get a conceptual understanding of rounding. Before Common Core, I would only use the steps of rounding, but now I am aware of the importance of the students actually understanding why they do what they do.

I ask a question. "Who knows what it means to round?" One student says, "When you round to the biggest number." Another student answers, "A number that is closest to the actual answer." I let the students know that the number line can give them a visual of what rounding means.

I write 24 x 87 on the board. I draw 2 number lines on the board. I ask the students, how do we round 24. The students go back to the rules of rounding. I ask, "What 2 numbers are we deciding to round 24 to?" The students knew that the number would either be 20 or 30. I put those two numbers on the number line, one on the left end, the other on the right. Then, I explained to the students that before we put the number 24 on the number line, we need to find the halfway mark on the numberline. This will help us put the other numbers on the number line. After, we but 25 on the number line, the students were able to see where to put the number 24. Then I asked, "Is 24 closer to 20 or 30?" The students all agreed that 24 was closer to 20.

We repeated the process with 87. The students decided that 87 was closer to 90. So our multiplication problem was 20 x 90. The students used the short cut that they learned in a previous lesson to multiply this problem. They multiplied the 2 basic facts of 2 x 9 = 18, then added 2 zeros to get 1,800. (We have discussed the expanded algorithm. However, with there being zeros, it is easier for the students to multiply the basic facts and add the zeros. They are aware of the value of each digit. For this lesson, I am concentrating on them understanding the concept of rounding.)

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#### Group or Partner Activity

*20 min*

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 (**MP3**).

For this activity, I put the students in pairs. I give each group the Estimate Products group activity sheet. The students must work together to place the numbers on the number line** (MP4)**, then estimate to find the product. They must communicate precisely to others within their groups **(MP6)**. 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 **(MP3)**. From the video, you can hear the students discuss 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 **(MP6).** 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 numbers should you have on the number line? Why?

2. How did you come up with your answer?

3. How does the number line help you?

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.

From this sample of Student Work, you can see that for pair did an excellent job of putting the numbers on the number line and estimating the answer.

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.mathsisfun.com/numbers/estimation-game.php

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#### Closure

*15 min*

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.

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- UNIT 1: Fractions
- UNIT 2: Skills Review
- UNIT 3: Algebra
- UNIT 4: Geometry
- UNIT 5: Patterns & Expressions
- UNIT 6: Problem-Solving Strategies
- UNIT 7: Decimals
- UNIT 8: Measurement and Data
- UNIT 9: Multiplication and Division Meanings
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- UNIT 12: Multiplying and Dividing

- LESSON 1: Multiplying by Multiples of 10 and 100
- LESSON 2: Using Rounding to Estimate
- LESSON 3: Is Your Answer Reasonable?
- LESSON 4: Using Clues to Multiply or Divide
- LESSON 5: Using Mental Math to Multiply 2-Digit Numbers (Are You My Match?)
- LESSON 6: Multiplying 2-Digit Numbers by Multiples of Ten
- LESSON 7: Multiplying Greater Numbers
- LESSON 8: Modeling: Multiplying a 2-digit number by a 1-digit number
- LESSON 9: Multiplying 2-digit number by 1-digit number
- LESSON 10: Multiplying a 3-digit number by a 1-digit number
- LESSON 11: Estimating Products
- LESSON 12: Multiplying 2-Digit by 2-Digit Numbers
- LESSON 13: Multiplication: Arrays and an Expanded Algorithm
- LESSON 14: Multiplication Unit Assessment
- LESSON 15: Using Mental Math to Divide
- LESSON 16: Estimating Quotients
- LESSON 17: Dividing with Remainders
- LESSON 18: Dividing 2-Digit by 1-Digit Numbers
- LESSON 19: Dividing 3-Digit by 1-Digit Numbers
- LESSON 20: Deciding Where to Start Dividing
- LESSON 21: How Much Will They Get?
- LESSON 22: Factors
- LESSON 23: Prime and Composite Numbers