##
* *Reflection: Developing a Conceptual Understanding
A Concrete Look at Long Division! - Section 4: Activity

We are taking it back to the old school ladies and gentleman. For so long we were able to count on the calculator to do the dirty work. Common Core is taking the calculator out of the hands of students. With this, in 7th grade we are asked to teach long division again. There is a purpose for conceptual understanding with long division as well as using the algorithm. I feel it is purposeful for students to understand why long division works because one they are given real world problems they are going to need a deeper understanding of grouping totals. I start with conceptual understanding and move into the algorithm with the next lesson. 1 inch connector blocks are a great hands on resource to use.

*Building a conceptual understanding.*

*Developing a Conceptual Understanding: Building a conceptual understanding.*

# A Concrete Look at Long Division!

Lesson 8 of 12

## Objective: SWBAT model long division with 1, 2, and 3 digit whole number dividends, and apply the algorithm to convert fractions to decimals.

## Big Idea: This lesson is multi-layered. It offers the students an opportunity to understand long division, then transitions into the written form using the algorithm once understanding is evident. Finally, the students will apply the algorithm to convert fractions t

*42 minutes*

**Why is this lesson awesome? **This lesson is multi-layered. It offers the students an opportunity to understand long division, then transitions into the written form using the algorithm once understanding is evident. Finally, the students will apply the algorithm to convert fractions to decimals.

#### Resources

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#### Teacher Guided Notes

*1 min*

**CCSS:** 7.NS.2d

**Teacher Guided Notes: **For so long so many of us have taught Divide, Multiply, Subtract, and Bring Down. This is necessary for long division, please make no mistake, however how do we know our students truly understand DIVISION? Let’s look at what has been going wrong for decades.

**Where we go wrong:**

Let's begin with the question 73 divided by 3.

Think about the 'confusing' steps involved in this question:

1. How many times does 3 go into 7?

(It's really not a 7 is it? It's 7 10's which means it's a 70.)

2. Put a 2 on the top.

*(Why? Answers previously have been put to the left or on the bottom.)*

3. What is 2 x 3?

*(Multiply? Isn't this a division question?)*

3. Now, put a 6 under the 7 and subtract 6 from 7.

*(What? Now the answer goes on the bottom and then subtract?)*

3. Bring down the 3.

*(Another step to remember that doesn't quite make sense.)*

4. How many times does 3 go into 13?

*(All this to determine how many times 3 goes into 73?)*

5. Put your answer on the top.

*(Answers on the top, on the bottom, how does all of this get remembered?)*

6. Subtract 12 from 13.

*(Subtraction again? But this is a division question!)*

**The above process is too confusing to a child. They can't remember the steps involved and therefore find long division completely confusion and lacking any sense. Typically, the child says 'what do I do next' because they lack the understanding**.

**A Different Approach to teaching Long Division:**

We need to get concrete to ensure that the process is understood. We will need strips for 10's and small squares for 1's. Just like you use buttons or counters for addition and subtraction.

Put the question into an authentic context, something like: "There are 73 pieces of fudge to be shared by 3 people"

Ask the child to 'represent' or build' the number 73. It will look something like this:

|||||||**...** (7 tens to represent 70 and 3 dots to represent 3)

Now ask the child to physically begin sharing into 3 groups.

1 Group of 10 will be left out which means the group will need to be exchanged for 10 ones. The child now has 13 ones to divide and will see that 1 one is left over which becomes the remainder.

|| |
|| |
|| |

**..****.** Remainder of 1

The child then has a complete visual of what 73 divided into 3 groups looks like.

The child counts how many are in a group and states 24 with a remainder of 1.

**Why does this work?** Because the child has visually seen what it looks like to divide. Many of experiences with this concrete method of dividing will eventually lead them to understand the actual algorithm above and then be able to use it which is exactly how they learned to add and subtract - concretely first!

Therefore here are the steps to master long division:

1. Use tens and ones (strips or base 10 pieces work fine) to model the question and answer.

2. When understanding is evident, move to the written form. Allow the student to use notations beside the division question to show the 10's and 1's, circling the groups as they perform.

A great video resource to watch before you teach this lesson is http://www.youtube.com/watch?v=DYJz5LofFSE this 13 minute YouTube video instructs how to use what they call the **fair share method**, and repeated subtraction. The **fair share** method is what is described above. **I highly recommend you watch the video before doing the bell ringer with your students**.

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#### Bell Ringer

*10 min*

**Bell Ringer**: Do you know your VOCAB?

Students will enter the room and be given the bell ringer as they enter the room. My students line up in the hall before entering the room, as they enter the room, they are greeted with a hello, and bell ringer. Students know the routine is to sit in their assigned seat and get started right away.

This bell ringer is a quick formative assessment over long division vocabulary. The students are instructed to create a long division equation that has a 3 digit dividend, a single digit divisor, and the quotient must have no remainders**. **

Students will have 5 minutes to complete this bell ringer. Having students color the terms in the equations allows you to quickly walk the room and identify those students who may need reinforcement. If there are a group of students who have no idea what to do, I would lead them to the quizlet.com resource for a quick tutorial.** **http://quizlet.com/1823088/long-division-vocabulary-flash-cards/ this online resource is a self-explanatory site that students can use with little to no help. They may opt to print out the flashcard in which they can glue into their Interactive Math vocabulary books**. **

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#### Activity

*20 min*

**Activity: **Do we Understand Division, or Know the steps? Well, why not both?

You may opt to do this activity as individual students, pairs, or groups. When I do this activity I like to model an example first. However, to have students truly immerse themselves into **MP1 **it would be great to give each student or group the manipulatives needed for the activity, an equation, and assess how they will use the manipulatives to solve the equation. If you opt to do this, the directions should be clear. Each student or group must model using their strips or base ten blocks. Each step should be illustrated and explained. If you choose to model an example for your students first, it would still be helpful to illustrate each step and explain each illustration.

Have each student complete 10 division equations in class using the fair share method.

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#### Exit Ticket/Homework

*10 min*

**Exit ticket/Homework**: 10 minutes before the bell rings, have students clean up, and complete the exit ticket. Anything not completed in class, please have the students complete for homework. Each student will complete the **Exit ticket**.

Students will complete the exit ticket and turn it in to you before they line up. This is the exit out of the room.

#### Resources

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- UNIT 2: Integers
- UNIT 3: Proportional Reasoning with Percents
- UNIT 4: Proportional Relationships
- UNIT 5: Proportional Reasoning
- UNIT 6: Rational Numbers
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- LESSON 1: Pre-Assessment Adding and Subtracting Signed Rational Numbers
- LESSON 2: Post Assessment Adding and Subtracting Rational Numbers
- LESSON 3: Determining situations in which opposite quantities make A FUN LESSON!
- LESSON 4: Now let's Demonstrate! Can I prove mastery of additive inverse on a number line?
- LESSON 5: Determine the Distance Between Two Rational Numbers on a Number Line
- LESSON 6: Now Let's Apply the Knowledge Gained! Determining the distance between two rational numbers on a number line.
- LESSON 7: Am I Terminating or Repeating?
- LESSON 8: A Concrete Look at Long Division!
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- LESSON 10: The World Of Inverse Operations!
- LESSON 11: Pre-Assessment Multiplying and Dividing Rational Numbers
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