How Close Is "Close"?

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Objective

SWBAT: • Compare decimals and fractions • Place decimals and fractions on a number line • Generate fractions and decimals that fall between two numbers

Big Idea

How close is “close”? Students generate fractions and decimals that fall between “close” numbers and are confronted with the density of rational numbers.

Do Now

10 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.  Here I want students to determine what the given amount is closest to: 0, ½, 1, 1 ½, or 2.  I am wondering what students do with three-twelfths.  Some students may recognize that it is equivalent to ¼, which is exactly between 0 and ½.  I am also interested to see what students think about 8/5.  Do students realize that it is greater than 1?  Do students recognize that it is equivalent to 1 3/5 or 1.6? 

I ask a couple students to share out their strategies for problem 2 and 4.  I call on students to share if they agree or disagree with the students’ ideas.  If the students don’t use it, I show how we can use the number line to help us.

 

How do they compare?

15 minutes

I have students work with a partner on this activity.  They did the same activity in the previous lesson, except today’s lesson includes numbers larger than 1.  I walk around and ask students to explain why they have chosen a particular sign.  I am looking to see what strategies students are using to compare decimals with decimals and fractions with decimals.   Some common mistakes are students mistaking 1.07 and 1.7 as equivalent and thinking 1.250 is greater than 1 ¼. 

After about 3-5 minutes, we come back together as a class.  I declare that for #4 I believe they are equal.  I ask for a show of hands, to see whether people agree or disagree with me.  I have students share out their ideas.  Students are engaging with MP3: Construct viable arguments and critique the reasoning of others.  I ask for volunteers to explain their thinking about problems 2 and 3.  These are questions that are sometimes a struggle with students. 

Next I ask a few questions about the number line.  An easy way to make this a movement break is to set up three labels along a classroom wall: 0, ½, 1, 1 ½ , and 2.  I ask a question and students have to go and stand by the number that they believe is closest. 

I ask:

  • Is 1.4 closer to 0, ½, or 1, 1 ½, or 2?
  • Is 1.07 closer to 0, ½, or 1?
  • Is 1 ¼ closer to 0, ½, or 1?

After each question I briefly ask students why they are standing in a particular place.  I am interested to see where students stand for 1 ¼, since it is exactly in between 1 and 1 ½.

How close is "close"?

25 minutes

Note:

  • For the previous lesson (Where does it fall on a number line?), I used the pre-test data to Create Homogeneous Groups.  I will have students working in the same groups of 2-3 students.

I have students move to their groups.  I explain that students will be choosing close numbers. I ask students to raise their hands and share out what they think it means for two numbers to be “close”.  I write these ideas on the board – we will return to them in the lesson closure.

 I pass out a Group Work Rubric for each group.  Students are engaging in MP6: Attend to precision, MP7: Look for and make use of structure and MP8: Look for and express regularity in repeated reasoning.

As students are working, I walk around and monitor student progress and behavior.  I am looking to see how groups are deciding on their close numbers.  Are most groups choosing close decimals for close fractions? What strategies are students using to find fractions between the numbers?  Are they using common denominators?  How are students generating decimals?  Are they extending the decimal places?

If students are struggling, I may ask the following questions:

  • What close numbers has your decided on?  Why?
  • What is a fraction that you think might fall between those two numbers?  Why do you think that?  How could you prove it?
  • What is a decimal that you think might fall between those two numbers?  Why do you think that?  How could you prove it?

If students are successfully completing their work I give them a new task.  They must choose two of the fractions or decimals that fall between their “close” numbers and they must go through the activity again (generating numbers between them, ordering them, and creating a number line).

Closure

10 minutes

 For Closure I return to the question, “What does it mean for two numbers to be close?”  I read the notes from our previous discussion.  I ask students to raise their hands and add their thoughts now that they’ve spent more time on it.  I am looking for students to realize that close is a relative term when it comes to comparing numbers.  Two numbers may appear close to one another (perhaps because of the way they are displayed on a number line), but really there are many numbers that fall between those two “close” numbers.    

I ask students to share out strategies that their group used to generate fractions and decimals between two numbers.    Then I ask one group to share out their starting “close” numbers.  I ask the group how many fractions and decimals fall between their numbers.  Some students may offer a number.  Other students may say that they don’t know.  I pose the same question to the class.  I want students to understand that there are an infinite number of decimals and fractions that fall between just those 2 numbers.  I show them the lesson image about 5.223. We could continue to find two numbers between the numbers, and then find two numbers that fall between those numbers, and so on. 

Instead of a ticket to go, I collect the students’ work to look at.