Rational Numbers and Integer Practice

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Objective

SWBAT: • Define rational numbers • Place rational numbers on a number line • Compare rational numbers • Add integers

Big Idea

What is a rational number? Which is greater: -1.6 or -1.5? Students work to answer these questions while also practicing adding integers.

Do Now

7 minutes

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’s problem is very similar to the do now in the previous lesson, except the numbers are larger.  Some students may draw a number line.  Other students may draw a picture of counters. 

I ask for a volunteer to share out his/her thinking and answer.  I ask the class if they agree or disagree and why.  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 ask for a student to explain an overdraft fee.  What if there is an overdraft fee of $15, what will be Bianca’s new balance?

Number Categories

3 minutes

We return to this graphic. I ask students, “What is special about rational numbers?” and “What is the difference between integers and rational numbers?”  I wan students to recognize that rational numbers are integers and numbers that can be expressed as a fraction, whereas integers are positive and negative whole numbers.

 

Number Line Review

10 minutes

Note:

  • Before this lesson I use the ticket to go data from the two previous lessons (Adding and Subtracting Integers on a Number Line and Adding and Subtracting Integers with Counters) to Create Homogeneous Groups.  Students will work in groups of 2.

We review that numbers on a horizontal number line increase from left to right.  I ask students what numbers are represented in problem 1 and 2.  A common mistake is that students think number 1 represents 1 ¾.  I remind students that we can see how many hops are between 1 and 2.  If we start at 1 and hop to 2, it takes 3 hops, therefore each tick mark represents 1/3.  The dot represents 1 2/3.  I ask students how I would represent 1 2/3 as a decimal.

I have students try problem 2 on their own.  I ask, “What does each tick mark represent?  How do you know?” and “What decimal represents that number? “

For problem 3, I give students some rational numbers to plot on the number line.

Possible Rational Numbers:

  • 0.5 and -0.5
  • -1 ½ and  1 ½
  • -3.5 and -4.5
  • 2 3/4 and -2 ¾

A common mistake is students confuse where rational numbers like -1 ½ go, in relation to -1.  Some students may place -1 ½ mistakenly where -1/2 is on the number line.  If this happens I remind students that they can think of a number line as a reflection around 0.  -1 ½ falls exactly in between -1 and -2.  I ask, “Is -1 ½ greater or less than -1?”  Since -1 ½ is less than -1, it should be to the left of -1.

Students may have similar struggles with -2 ¾.  I ask students similar questions about the placement of -2 ¾. 

I have students complete the comparing problems independently.  After a few minutes, I have a student come to the document camera to show and explain their thinking for each problem.  For problem 2, I have students draw a number line that goes from -3 to -4 and we place -3.6 and -3.2 on it to help us compare it.  A common mistake is that students ignore the negative sign and compare the numbers as if they were positive.  I require students use the number line to explain and justify their answers (MP4).

Comparing Rational Numbers Dice Game

15 minutes

Note:

  • Each partner pair will need a 1-6 die.  If I want to make it more challenging for partners, I give them a 1-9 die.

I review the rules and do an example with the class.  Students work with their partners to play the game.  Students are engaging with MP4: Model with mathematics.  Students are more engaged any time I make a game out of math practice.  I walk around and monitor student progress.  Some students struggle with the number line.  If this is an issue, I ask students what their tick marks should count by.  I also ensure that students are not changing the placement of their digits once their roll is finished.

If there are common mistakes, I note them.  With a couple minutes left in this section I present the mistake and ask students to tell me if they agree or disagree and why.

I collect the students’ game worksheets so I can look over them after class.

Integer War

15 minutes

Notes:

  • Each partner pair will need a full deck of cards.
  • Have bags of counters ready for each student if they choose to use them.
  • I use the ticket to go data from the 2 previous lessons to determine if there are students who continue to struggle adding integers.  If there is a small group of these students I will pull them to work with me.

I review the rules and go over the examples together.  I review expectations and stress that students have blank numbers lines they can use and yellow and red counters if they prefer. 

If I pull a small group of struggling students, here is how I may structure the activity:

  • Instead of competing with each other, we will use the game as whiteboard practice.
  • Two students will flip one card each.
  • I ask students to tell me what number each card represents.
  • Then each student will work independently (with any models they choose) to add the two integers together. 
  • When they finish, they show me their work on the whiteboard.  If it is correct, I give them another problem.  If it incorrect, I tell them where they made a mistake.
  • Two more students flip one card each and we repeat the process.

If students need an additional challenge they can play “Guess Their Number”.  This requires students to work backwards to determine the value of their own card. They can also choose to work on the Magic Square Challenge with integers.

Closure and Ticket to Go

10 minutes

 For Closure I ask students to define rational numbers.  I draw a quick number line that goes from -4 to -3.  Where is -3.8?  I have students share their ideas for placing -3.8 on the number line.  Then I ask students which is greater: -3.8 or -3.6?  I also ask students what strategies they had when they were playing integer war.  If students don’t mention it, I talk about the “Make Zero” strategy.  For example, lets say I have to add -6 and 8.  I know that if I add 6 to -6 I get 0, since they are opposites.  So I break the problem up like this in my head: -6 + 8 = (-6 + 6) +2 = 2.  This mental math strategy allows me to add integers in my head.

I pass out the Ticket to Go for students to complete independently.  At the end of class I pass out the HW Rational Numbers and Integer Practice.