##
* *Reflection: Developing a Conceptual Understanding
Making sense of Remainders - Day 1 - Section 3: Concept Development

As you can see in my reflection video, this was a great lesson. Students are truly understanding the concept of remainders and worked at investigating sharing and dropping remainders.

I liked how my students were discussing math today in this activity. Math Practice Standard 3 was used extensively as students worked with their table groups to make sense of the problem, and convince each other, especially since most students at this point, do not know a procedural algorithm for division. Not knowing an algorithm allows them to talk with each other about strategies and the concept of division. Students don't get hung up on just showing their partner or table mates how to do a procedure. I find that since students don't know an algorithm, they talk and discuss the problem much more thoroughly.

*Making Sense of Remainders*

*Developing a Conceptual Understanding: Making Sense of Remainders*

# Making sense of Remainders - Day 1

Lesson 4 of 7

## Objective: SWBAT investigate real world problems in which remainders are shared or dropped.

#### Warm Up

*10 min*

I begin this warm up by reviewing area models to show division problems with remainders. I ask students to use graph paper to model 25 / 4, 36 / 5, and 24 / 7. This is challenging for my students to think about an area model to show division. After each problem, I have a student show his/her area model under the document camera. I am very explicit in pointing out the quotient and the remainder in each of the problems to reinforce division vocabulary.

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#### Concept Development

*40 min*

It often takes a jump in understanding for students to apply the procedural algorithm of division with remainders to real-world situations where remainders are encountered. A child who can easily calculate 40 divided by 6 = 6R4 will too often state 6R4 as the answer to the number of cars necessary to transport 40 children to a baseball game if 6 children can fit in each car.

The problems in this lesson will first review the concept of division as proportional reasoning involving equal shares and then they will lead children to discover the three usual ways of dealing with remainders in real life: they are either used to round up to the next whole number, they are dropped and discarded, or they are split evenly among the participants.

I begin this lesson by having students repeat after me. We chant; Round it, Drop it, Share it! I try to make this exciting by having students punch the air each time they say it, or clap their table at the same time.

Then, I explain that many times in life, people either round remainders, drop remainders, or share them. Students take a piece of paper and fold it into fourths. Then as table groups, students work through the problems in the division power point. Division!!!!.pptx

I give groups time to work and then have them stop after they have completed each problem. This power point, and stopping after each problem, takes the entire class period. We did not have time for the very last slide in which students were to write a word problem in which the remainder is reported. I assigned this as homework. We discussed the phrase that would be needed in the problem in order for the remainder to be reported. Students understood that they would need a phrase like, 'how many _______ are remaining?'

A management and engagement strategy I use in order to keep noise down, and students focused and on task while some groups finish and wait for other groups, is that I allow them to make a math sketch to go with their problem. For example, students might draw a candy bar sketch for the first problem. This allows them to keep focused on math, while allowing other groups the opportunity to finish the problem before we begin a class discussion.

Students work with Math Practice Standard 3 in this lesson. When students learn to defend their thinking and critique the reasoning of peers from a young age, they are more likely to maintain and improve this behavior in later years. Student discourse requires students to think about problems in a variety of ways and to defend their solutions against the critique of their peers. Student discourse promotes a higher level of thinking than merely stating answers or answering knowledge-based questions. Students can arrive at a “correct answer” but still not understand the concept; by defending their answer they may deepen their own understanding and demonstrate that understanding to the teacher. This process may enable classmates to gain a better understanding of the concept, confirmation of their own thinking, or an appreciation of other ways to arrive at the same answer.

The division poster in the resource section is hanging in my room for students to refer to throughout the year.

In this video, you can see a group solving a problem in which the remainder is dropped.

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- UNIT 1: Getting to Know You- First Days of School
- UNIT 2: Multiplication with Whole Numbers
- UNIT 3: Place Value
- UNIT 4: Understanding Division and Remainders
- UNIT 5: Operations with Fractions
- UNIT 6: Fraction Equivalents and Ordering Fractions
- UNIT 7: Division with Whole Numbers
- UNIT 8: Place value
- UNIT 9: Geometry
- UNIT 10: Measurment
- UNIT 11: Fractions and Decimals

- LESSON 1: A Remainder of One
- LESSON 2: Practicing Division with SumDog
- LESSON 3: The game of Left-Overs
- LESSON 4: Making sense of Remainders - Day 1
- LESSON 5: Making Sense of Remainders Day 2
- LESSON 6: Making Sense of Remainders Day 3
- LESSON 7: Lots of Laughs with Finding Unknown Factors