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# Lines Around the World: Combining and Graphing Integers on a Number Line

Lesson 9 of 20

## Objective: SWBAT combine two or more integers by writing numerical expressions and plotting them on a number line

## Big Idea: Students move from station to station individually or in partners graphing integer addition sentences and writing addition expressions

*40 minutes*

#### Do Now

*10 min*

Students enter the class and find a “Do Now – Pop Quiz” on their desk that will include 5 integer combination questions. The questions will assess student knowledge of combining integers with opposite signs and integers with the same sign(i.e. 16 + (-5) and -6 + (-8)). The pop quiz will help me generate a list of students still struggling with integer addition. Students who have not mastered this concept will be encouraged to work in groups so that I can work with them more closely. There are two forms of the quiz, A and B. The letters are indicated at the top right hand corner of each quiz. I will have answers ready to check as students finish.

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Students are instructed to take out their homework from the previous night. They are also instructed to fill out their heading in their notes and to copy the aim off the power point.

We begin by reviewing the answers to the homework assignment from the previous night. That assignment included 6 word problems. Students were responsible for writing and evaluating addition expressions to represent each word problem (i.e. a “plane descends 1200 feet and then ascends 500 feet”; Glencoe Workbook online, pg 18). I review the answers to the homework by asking students to read each word problem. After a student reads a portion that could be represented using an integer I ask them to stop and I show everyone which integer represents that part of the problem. Some students may ask about the relationship between adding the inverse and subtracting. I simply state that their number sentence gives the same result, but they must write an addition sentence because of the directions. We discuss the answers to each expression by thinking about the use of counters we explored the previous day. I stop when we get to the golf problem to explain the rules of golf. Many students do not understand the idea of “par”, so I make sure to make time to explain golf, how it is played, and the meaning of “par”. To check for understanding, I ask a volunteer to explain what “0 par” would mean, as well as “1 over par” and “1 under par”. I also point out that in golf you want to get a negative score or zero, not a positive score. Students are asked to explain why one would want a negative or zero score.

After the homework has been reviewed, we fill out the Cornell Notes. I review the position of negative and positive integers in relation to zero for both horizontal and vertical lines. Then, we complete two sample problems together: one that asks students to draw the “arrow annotation” given an expression and another that gives the arrow notation and asks students to write an expression. If there is time, I also show students how to draw arrow notation for combining two negatives. Student questions are answered if there are many, or if there are only a few, they are encouraged to ask their questions during the game.

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#### Task

*15 min*

I explain the structure of the game to students: it is called “Around the World” because they will be moving around the room to complete the problems at different stations. The first station includes 4 problems like the examples in the Cornell notes: two will ask them to draw the arrow notation on a number line and two will ask them to write expressions given number line arrow notations. The second station includes two word problems like the examples in the homework. Students will be writing expressions to illustrate the word problem. The third station includes one problem about combining integers along the number line, but one of the numbers is a variable. The last station is a puzzle station. Two puzzles are given, students are to choose one. If they answer correctly, they will receive a free homework pass. Additionally, students will receive achievement points for clearing stations. One achievement point will be awarded for completing each station if a student chooses to work independently. If a student chooses to work with a partner, they will each receive two achievement points after completing each station. **MP1** and **MP2** are being used to explore integer addition in these three different formats (skill level problems, world problems, algebra, and puzzles).

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#### Closing

*5 min*

When 5 minutes are left for class, students come back to their seats. They are instructed to use a sticky note to write one question they still have about combining integers and stick it on a piece of chart paper before exiting the room.

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*Responding to Jennifer Timpe*

Hi there, thank you for reviewing this lesson :)

Are you asking about the questions students posed during the closing? Unfortunately I did not take a picture of those at the time, nor did I keep them over the last two years.

From my experience and memory over the last few years teaching this lesson, lingering and reoccurring questions that come up often include:

- questions about double signs "what should I do when I see two negatives? how do you know what direction you should go in on the number lines when there are two signs"
- "i still don't understand when I am supposed to add and when I am supposed to subtract"

Please let me know if there's anything else I can provide.

One thing I WILL say about this topic is that students asking questions like those above need more examples with integer chips so that they can continue to explore the rules for combining rational numbers. The number line is also something very confusing to students because it is not concrete, like the integer chips, especially when it comes to those double signs. If students seem to be having a problem understanding concretely what is happening on those number lines, I like to refer to temperature problems, especially since they seem to be the focus in 7th grade ccss. Moreover, making the shift to ccss means students have to be pushed to make sense of operations, including those with double negatives. For example, in the problem 7 - (-6) we could relate to temperature by translating it as

it is seven degrees above zero. The **first negative** indicates a movement in the **opposite **direction of the **second negative sign (left/down)**. Theopposite of left/down is right/up. Therefore we travel right/up six degrees.

After this type of example, many students might ask, "but why 'up 6'? isn't 6 a negative?" which indicates that they are still not understanding that we have already dealt with that negative by giving it a direction.

This idea can also be explored with integer chips which can be manipulated.

- We have 7 reds (positive)
- we are being asked to take away (
**first negative sign**) - 6 blues (
**second negative sign**)

but since all we have is reds, we can't really take out any blues...

We could add 6 zero pairs. That's six pairs of blue/red chips. T**his is a great idea that students ought to discover on their own through magic teacher questioning and guidance. I call it magic because I struggle every year to find the right kind of questioning to lead my students there. **I'll be honest, sometimes I fall prey to the easy solution, and I just give them the rules. "If you see two negatives, make them positive". But that inevitably blows up in my face as students start following this rule incorrectly, for example, -5 - 7 = 5 + 7 = 12 ("You said when I see two negatives make them positive!") :(

This was a long response, I'll stop here. Please let me know if there's any other resource I can provide. :)

| 2 years ago | Reply

Could you please post the questions from all the stations? Thanks!

| 2 years ago | Reply##### Similar Lessons

###### Adding and Subtracting Integers on a Number Line

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- UNIT 1: Integers
- UNIT 2: Operations with Rational Numbers
- UNIT 3: Expressions and Equations - The Basics
- UNIT 4: Multi-step Equations, Inequalities, and Factoring
- UNIT 5: Ratios and Proportional Relationships
- UNIT 6: Percent Applications
- UNIT 7: Statistics and Probability
- UNIT 8: Test Prep
- UNIT 9: Geometry

- LESSON 1: The Numbers Game: Playing by the Rules
- LESSON 2: The Numbers Game: Playing by the Rules
- LESSON 3: Exponents: It's Gotta Be the Power of 3
- LESSON 4: Rolling with the Order of Operations
- LESSON 5: Bingopposites
- LESSON 6: Øriginal Distance: Absolute Value and Additive Inverse
- LESSON 7: Order Up! Ordering and Comparing Integers
- LESSON 8: All That and a Bag of Chips! Using Counters to Combine Integers
- LESSON 9: Lines Around the World: Combining and Graphing Integers on a Number Line
- LESSON 10: MAP it Out
- LESSON 11: What's up with that? The Connection Between Addition and Subtraction of Integers
- LESSON 12: Note the Arrows: Modeling Addition and Subtraction of Integers on Number Lines
- LESSON 13: Man Your Station! Adding and Subtracting Integers in Real World Situations
- LESSON 14: Lines and Patterns: Difference, Change, and Multiplication
- LESSON 15: We Are a Family!
- LESSON 16: Trashketball!
- LESSON 17: Quiz Day
- LESSON 18: Rewind! Reviewing Integer Operations and Critical Thinking
- LESSON 19: Unit 1 Test
- LESSON 20: Error Analysis: Unit 1 Test