Identifying Integers

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SWBAT identify integers.

Big Idea

Students understand the difference between integers and rational numbers.

Do Now

10 minutes

Today's Do Now is a set of Warmup problems that review concepts learned earlier in the course.

Do Now

Compare <, >, =

1) 0.003    0.0030

2) 4.05      4.0500

3)  2/3        8/12

After 5 minutes, I will call students forward to the board to show their answers and to explain their reasoning. I am interested in how my students will make these comparisons. Each problem offers students a different conceptual challenge. So, I will listen carefully as students explain their answers to make sure that they are referencing concepts in their explanations.

Introduction to Integers

10 minutes

If I were to Cold Call my students and ask, "What's an integer? When do you use them?" many would not be able to give me a good answer. So, I like to start off this lesson by watching a video from PBS's Math Club:

What is an Integer?

The video is a good introduction to integers using an place where they are used (opaquely) in students' everyday lives.

As my students watch the video I will make a list of four relatively recent movies on the board. For example, at the time that I wrote this lesson these four movies were well known to my students: 

1) Twilight

2) Frozen

3) Monsters University

4) The Croods

When the PBS Math Club Video ends I will randomly select 10 students to rate each movie with a thumbs up or thumbs down (Thumbs up/Thumbs down). Once the movies are rated, I will select a student to make a comparison statement about any two of the movies. I'll say something like, "Based on these ratings, which movie would you choose to see?"  I am hoping that a student will respond with a statement like, "I'd go see Twilight because -2 is greater than -8".  An answer like this often confuses some of my students. So, I will usually follow up with a question like, "Would you rather see a movie with 2 thumbs down or 8 thumbs down?"

We'll discuss several different statements until we reach a point were I feel my students are ready to think about a rating system like Youtube's as an application of Integers.

Digging Deeper into Integers

15 minutes

Following our informal exploration of integers using video ratings on Youtube, I will give my students a formal definition. I'll write on the board

Integers - the set of counting numbers, their opposites, and zero

To clarify the definition, I like to have a number line available for students to view. Depending on how the discussion of Youtube ratings went, I may ask, "Can anyone explain how this definition of integers relates our conversation of movie ratings?" I find it helps to anchor the definition in a context like this.

I also like to present weird counterexamples. I might say something like, "If I were counting the number of students in the class, would it be sensible to count like this:

1, 1 1/3, 1 2/3, 2, 2 1/3, 2 2/3, 3, .... ?

Once we've established the importance of Integers and Counting Numbers as sets, I will display a set of Real Numbers on the board. One at a time, I will call students to the board to identify a number in the set that is an Integer (see Identifying Integers Activity). As students make their selection, we will discuss why each number falls into each category, Integers. When we have identified all of the Integers in the set, I will point to the other numbers and ask students to volunteer to explain why they are not integers. 

After we have discussed all of the numbers on the board I like to talk about the phrase "opposite of an integer." I may erase all of the non-integers from the set of numbers on the board and ask students to identify the opposite of each Integer. Since we have been working as a class, I like to begin this activity by asking students to discuss the possibilities with a partner. Then, I will let volunteers share with the class.

To bring this section of the lesson to closure I like to have my students discuss the following questions with their partner:

  • Does every integer have an opposite?
  • What's the relationship of opposite integers and their distance from zero?

Independent Practice

10 minutes

To assess students' understanding of integers and opposites, I will post the following Independent Practice Problems on the board for students to complete in their notebooks.

Independent Practice

1)What is the opposite of -5,092 ?

2)What is the opposite of 0 ?

3)Simplify -(-9)

4)Simplify -(-12)

5)Simplify -[-(-15)]

6) Explain a real world situation in which integers are used.

7)  Identify the integers in the following list:

 -9,  3/7,  -6,  10/2,  0,  134, -2/5, 34, -17

After 10 minutes, we will review the answers as a class. For Problems 3 - 5, some of my students may be confused with the meaning of more than one negative sign.  For Problem 7, a common misconception for students at this point is that any number with a negative sign is an integer.