Interval Notation

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Objective

SWBAT express and interpret different methods for writing intervals.

Big Idea

Explore the different methods of writing intervals in describing domains, ranges and intervals describing graphs.

Unit Terminology and Notation

At the beginning of each unit I give students suggested homework the learning targets for the unit.  These documents tell students the concepts they will study and how they will be tested. It also gives students suggested homework that can be done.  I assign some problems from these suggested problems but do not grade all the problems.

In my district, literacy is an important focus across the disciplines. We have read Cris Tovani's book "I read it But I Don't Get It." I will use many of the strategies from this book throughout the school year. In my classroom, I interpret this to mean that students will read and interpret mathematics. This is a college prep class and students will need to be able to read from a textbook when they go to college. I spend time throughout the year helping students with strategies necessary for learning mathematics.

Bell Work

10 minutes

As students arrive they are to think about the following question

Take out a piece of paper and define the mathematical terms domain and range.  Explain any words that mean the same thing.

I will use the think-pair-share (or in my case group) strategy to assess what students remember from prior courses. I ask students to share their thoughts with their groups. By sharing what they remember students help remind each other of material already learned.  I ask questions to help students connect our work to prior ideas in science and math.  Questions include:

  • What is the domain called in science?
  • What did you call the domain in elementary school?
  • Is "x" always the domain variable?

After groups work for 2-3 minutes, we come back as a class.  I randomly choose students from to share what was discussed in the groups about domain.

I follow the same instructional strategy for the range.

We now compare students' work with the textbook definitions. Some students want to find the definitions on the internet. We do this quickly by using smart phones and reading what is found. Even though these definitions are known to students seeing how the definitions are written in a textbook helps students begin to read mathematics.

I informally assess student's understanding by giving two examples. The second example is interesting since the rate is not the normal miles per gallon.

 

Review of Basic Functions

10 minutes

I again use background knowledge in discussing the different ways to write a domain and a range.  I want students to realize how much they already know.  By discussing the functions they learned in Algebra 1 and Algebra 2, I can determine what material the students have learned and what I need to reteach. 

I begin this section with the following questions:

What is the domain of a quadratic, cubic, linear? How do you write out the domain? As the students discuss I document the graphs  along with any information given about the function. Some students need to see the information and others need to write the information down to remember.

Do all functions have domains that are all real numbers?

Can you remember any that do not?

If students do not remember functions, I bring bring up the square root function.  I share graphs (page 2 and 3) of the square root and an exponential function and determine the domain and range for these functions.

 

How to Write Intervals

20 minutes

When students identify the domain and range of several functions, I am able to assess what notation they used and remember from Algebra 2. Most students remember the infinity sign and the notation for all real numbers. Once I determine what they remember, I  focus on what the students do not know or remember.

I hand out the interval notation notes. Reading mathematics requires an understanding mathematical "shorthand." I always say that mathematicians do not like to write a lot, so they have symbols that make writing quicker. These symbols are one reason why some people think math is hard.  

Students are given about a minute to scan the first page of the notes. I ask "have seen any these notations before? When and what do they mean?"

I continue looking over the first page of the notes. We  look at the first row. I explain that every notation is saying the same thing depending on the situation it may be written in a different format. We start by looking at {x:x>-2}.  I write out the meaning of this notation.( x such that x is greater than -2) We discuss how the the notation tells the variable and then puts a restriction on the variable. We look at the next box and I ask what the parenthesis means for the interval. We refer back to the first box to see how -2 is not included in the interval. Finally we look at the graph notice that there are 2 different ways to show the interval.

We continue working through the notes. As the class reads through the notes I ask some of these questions:

  • What is the difference between ( and [ or ) and ]?
  • What does the U mean?
  • What notation is put before negative infinity or after positive infinity?

At this point in the school year I do spend a lot of time on reading mathematics. As the year progresses the students are able read independently and then ask specific questions about the text.

 

 

Check for Understanding and Practice

10 minutes

Once the discussion of the notes is completed, I give students some examples. Students use individual whiteboards to record their answer. I move around the room to verify students are understanding. I have student share their whiteboard with the class and see if there are other answers. When there are more than one response we discuss which is correct and why. This is a great time for students to analyze other students work and explain others reasoning.

I assign interval notation practice and allow 3 class days to turn it in on time.

Even though the assignment is short I always give students multiple days to do homework. Giving extra days allows students a chance to get help if they do not understand the work.  

 

Closure

10 minutes

With a few minutes left in class I ask the following questions:

Which format is the easiest to understand?

Which format is the most confusing?

What confusions could happen with each notation?  

I always have the students look at how the an open interval in interval notation form can look like an ordered pair. 

We develop a table for the confusions of each notation.