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# Adding and Subtracting Integers with Counters

Lesson 6 of 17

## Objective: SWBAT: • Define and identify opposites • Add and subtract integers using counters

## Big Idea: What are opposites and what happens when you add them together? How can you use counters to model 7-9? What about 3 – (-3)? Students work with counters to model adding and subtracting integers.

*60 minutes*

#### Do Now

*5 min*

See my **Do Now** in my Strategy folder that explains my beginning of class routines.

Often, I create do nows that have problems that connect to the task that students will be working on that day. Today I want students to apply what they did in the previous lesson (Adding and Subtracting Integers on a Number Line) to a problem with larger numbers. Some students may draw a number line, while other students may draw a picture.

I ask for a volunteer to share out his/her thinking and answer. I ask the class if they agree or disagree and why **(MP3)**. I want students to recognize that if you spend more than you have in your account, you will have a negative amount of money à you owe the bank.

I briefly explain that when this happens it is called an “overdraft” and the bank actually charges you a fee for spending more than you have. Let’s say Bianca’s bank has an overdraft fee of $25. After the bank charges Bianca the overdraft fee, what is her new balance? How much money will it take for her to get back to $0?

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#### The Battle of the Integers

*10 min*

Notes:

- Each student will need about 20 two-sided counters. I use red/yellow counters.
- Timon Piccini created a video about the Battle of Integers. He uses red army men for negatives and blue army men for positives: Battle of Integers. The first couple problems in each section (Integer Addition and Integer Subtraction) correspond to the video.

I tell students that today integers are going to battle one another. Using two-sided counters is another way, in addition to using a number line, to model adding and subtracting integers.

I show students how they will draw pictures to show what they do with their counters. I explain that if we add 1 + (-1), the yellow and red soldier neutralize each other, we are left with 0. We do problem 1 together, then students work with their counters independently. I am checking to make sure that students make their model, draw their model, and write their answer. I do not expect students to draw a picture for problem 5, but I’m looking to see if they can recognize a pattern.

After 3 minutes, I stop students and ask what they notice. Some students may share that they all have the same answer, 0. Why is that? I want students to realize that if you add two numbers that have the same digit(s) but different signs, the result will be 0. I ask students what if I started with *n *counters, what would I have to add to *n *in order to get an answer of 0. Students participate in a **Think Pair Share. **I want students to generalize that *n *+ (-*n*) = 0. Students are engaging in **MP8: Look for and express regularity in repeated reasoning.**

We move on to the next page, “Opposites”. I review the definition that they developed. I have students raise their hands to share out the opposites to the integers listed. What is the opposite of 0? I want students to recognize that 0 is its own opposite.

Then I introduce the phrase “zero pair”. We call any two numbers that combine to make 0 a zero pair. Zero pairs are opposites because when you add them together you get zero.

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#### Integer Addition

*10 min*

We start by adding 3 positive yellow soldiers and 3 positive yellow soldiers to get 6 positive yellow soldiers. I model how I want students to draw the counters and students create their own model and drawing.

We move onto number two. We start with 6 positive yellow soldiers and we add 4 negative red soldiers (see my picture **6 + (-4) **in the resource section). It is important that students organize their counters into zero pairs here. We see that there are 4 zero pairs, where 4 positive yellow soldiers have neutralized 4 negative red soldiers. The result is that we are left with 2 positive yellow soldiers or 2. The positive yellow soldiers are victorious! We do problem 3 together and then students work independently on problems 4-8. Students are engaging with **MP4: Model with mathematics.**

I am walking around and monitoring student work. I am making sure that students are creating models and drawing their model for each problem. Some students may say, “I just know it is (-2)”. I tell them that today we are practicing with the counters and that they must use them to create a model.

Some students may catch on with the counters quickly, other students may struggle with the counters. If students are struggling I ask them:

- What soldiers are you starting with?
- What are you adding to this amount?
- How can you organize your soldiers to see if you have any zero pairs?
- Who wins? How do you know?
- How can we draw a model of your counters?

If students successfully complete their work, they can work on the Magic Square Challenge.

With a few minutes left, I have students come to the front to show their models of problems 5 and 8. I ask students if they agree or disagree and why. I ask students if they noticed any patterns while they were working on these problems. Some students may say they notice that when you add 2 negative numbers together, the result is a negative number. Other students may try to make rules about adding positive and negative numbers together.

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#### Integer Subtraction

*15 min*

I introduce the idea of surrender. In our battle, when we subtract an integer we are surrendering, or taking away those soldiers. We do the first example together. We start with 3 positive yellow soldiers. We need to take away 3 negative red soldiers. Here’s the problem: we don’t have any red negative soldiers to take away. I ask students if they have any ideas about what we could do to our counters in order to have 3 negative red soldiers to take away. A student may say to just add 3 negative red soldiers. I do this and show students that this does not represent the problem because if we have 3 positive yellow soldiers and 3 negative red soldiers we have 0, not 3 positive yellow soldiers. Another student may suggest adding zero pairs. If this doesn’t come up, I show students how we can add 3 zero pairs. This has *not *changed our problem, because we still have 3 positive yellow soldiers and 3 zero pairs. Now we can take away 3 negative red soldiers. What are we left with? Our result is 6 positive yellow soldiers (see my picture **3 - (-3) **in the resource section).

Students may struggle with this and that is okay! I mention that this was the football situation where the Negative Red Team got the ball, but was then pushed back. I reassure students to hang with it, we will be working on lots of examples and they will get it!

We work on problem 2 together. We start with three positive yellow soldiers. They have to surrender or take away 5 positive yellow soldiers. How will we do this? Again, some students may know that the answer is (-2) but not be able to model it with the counters. How can we use zero pairs to help us? How many zero pairs do we need to add so that they can surrender 5 positive yellow counters? After the surrender, what are we left with? We are left with 2 negative red soldiers, so our answer is (-2).

We do problem 3 together and then students work on problems 4-8 on their own.

If students are struggling I ask them:

- What soldiers are you starting with?
- What are you taking away?
- How can you use zero pairs to help you?
- After the surrender, what are you left with?

If students successfully complete their work, they can work on the Magic Square Challenge.

When a few minutes left, I have students come to the front to show their models of problems 6 and 8. I ask students if they agree or disagree and why.

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#### Mixed Practice

*10 min*

I have students work independently on these problems.

If students struggle, I ask them the questions I asked them in the previous sections. I may also pull a small group of students to work with.

If students successfully complete the problems, they can work on the challenge. I may also give them more challenging problems to see if they can apply what they know.

Possible problems:

- (-25) + 75
- (-123) + (-38)
- 35 – (-10)
- (-50) – 12
- (-87) – (-25)

#### Resources

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#### Closure and Ticket to Go

*10 min*

For **Closure **I ask what they notice about the mixed practice problems. I want students to notice that problems 4 and 5 as well as problems 6 and 8 have the same answers. Why is that? I want students to recognize that adding a number is the same as subtracting the *opposite* of that number.

I pass out the **Ticket to Go **for students to complete independently. At the end of class I pass out the **HW Adding and Subtracting Integers with Counters.**

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*These are very good lessons. I had so many ways of teaching integers even my head was becoming confused. Thanks for sharing | one year ago | Reply*

Andrea, thank you so much for sharing! Your videos are also very insightful as I am a first year teacher and still trying to find routines and transitions that work best for me and my students.

| one year ago | Reply

Andrea, thank you so much for sharing! Your videos are also very insightful as I am a first year teacher and still trying to find routines and transitions that work best for me and my students.

| one year ago | Reply

Andrea, thank you so much for sharing! Your videos are also very insightful as I am a first year teacher and still trying to find routines and transitions that work best for me and my students.

| one year ago | Reply

*Responding to Andrea Palmer*

Hey Andrea, thanks so much for the response! I totally get the zero pairs - That's how I taught it last year (the way you described in your comment). My question though is - why do you explain 3 - 5 using the zero pairs strategy, instead of explaining that I could just do 3 + (-5), which in counters would be 3 yellows and 5 reds... leaving us with 2 reds. Either way you get the same answer. I'm just wondering if that makes more sense to students that using the additional "zero pairs" strategy.

| 2 years ago | Reply*Responding to Jacqueline Brown*

Hi Jacqueline! Here's how I explain 3-5. We have 3 positives, but we need to take away 5 positives. In order to have 5 yellows, we need to add two zero pairs. Students need to be able to explain why adding zero pairs to the problem do not change our starting value. Since we have 5 yellows and 2 reds, the starting value is still 3. Then the problem tells us to take way 5 yellows. We are left with two reds, or negative two. It might help to also make the connection to back accounts here. If you have 3 dollars in your bank account, but you spend 5 dollars your new balance will be -2 dollars. In this situation the zero pairs were added by the bank and after spending 5 dollars you owe the bank 2 dollars. Does that help at all? Let me know.

Thanks! Andrea

| 2 years ago | Reply

For 3 - 5 in Integer Subtraction - what is your explanation for doing 3 - 5, instead of changing it to 3 + (-5)? My coteacher and I are having a disagreement about which one makes more sense to students :) Does it make sense to use zero pairs, if we can just add 5 red counters instead?

I used this lesson last year, and it was great, but I'm not sure which explanation makes more sense to a 6th grader. Thanks!

| 2 years ago | Reply*expand comments*

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- UNIT 1: Intro to 6th Grade Math & Number Characteristics
- UNIT 2: The College Project - Working with Decimals
- UNIT 3: Integers and Rational Numbers
- UNIT 4: Fraction Operations
- UNIT 5: Proportional Reasoning: Ratios and Rates
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- UNIT 7: Geometry
- UNIT 8: Geometry
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- UNIT 10: Review Unit

- LESSON 1: Pre Test
- LESSON 2: Where Does That Fall On The Number Line?
- LESSON 3: How Close Is "Close"?
- LESSON 4: What Are Integers?
- LESSON 5: Adding and Subtracting Integers on a Number Line
- LESSON 6: Adding and Subtracting Integers with Counters
- LESSON 7: Rational Numbers and Integer Practice
- LESSON 8: Show What You Know About Integers and Rational Numbers
- LESSON 9: Absolute Value and Stocks
- LESSON 10: Tracking Stocks and the Coordinate Plane
- LESSON 11: Tracking Stocks and Distance on the Coordinate Plane
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