##
* *Reflection: Diverse Entry Points
How Close Is "Close"? - Section 3: How close is "close"?

During group work, students were in homogeneous groups based on the ticket to go from the previous lesson. Different groups chose different numbers that they believed to be “close”. Here is a range of work from a number of groups.

This group was a group that struggled with the previous lesson. They chose these numbers and said they were close because they were “both between 0 and 1”. They only produced a few fractions and decimals, but the work shows that the students are able to change a fraction to a decimal. One issue is that they listed that 1/50 was equivalent to 0.2. I suspect that they confused 1/5 and 1/50. They are also able to simplify fractions. I watched this group work and their strategy was to come up with the decimal first and then change it into a fraction.

This group was one of the few groups that chose two “close” fractions, most groups chose to start with decimals. This group moved from tenths and used hundredths to find fractions that were between four-tenths and five-tenths. They list all of the hundredths between forty and fifty hundredths. This was an approach that many groups used. At the end of the second table, they have exhausted all of the possible hundredths. At this point, some groups said things like, “There aren’t any more fractions/decimals that fall between our numbers.” I told them to keep thinking. This group came up with 0.432. This took time, but ultimately they were able to see that there were indeed more fractions and decimals between their numbers.

This group picked two numbers that were further apart than most groups. Looking at their fractions, it is clear that they understand that they can create fractions that have different denominators than tenths and hundredths. They were one of the few groups to create a fraction out of 10,000. I did check in with the group to let them know that the lower table had to have values that were different than the top table. One issue is that the group listed 0.5 as falling between 0.5 and 1.

*Diverse Entry Points: Choosing "Close" Numbers and Generating Fractions and Decimals*

# How Close Is "Close"?

Lesson 3 of 17

## 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.

*60 minutes*

#### Do Now

*10 min*

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.

*expand content*

#### How do they compare?

*15 min*

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.

#### Resources

*expand content*

#### How close is "close"?

*25 min*

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).

*expand content*

#### Closure

*10 min*

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.

#### Resources

*expand content*

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- UNIT 1: Intro to 6th Grade Math & Number Characteristics
- UNIT 2: The College Project - Working with Decimals
- UNIT 3: Integers and Rational Numbers
- UNIT 4: Fraction Operations
- UNIT 5: Proportional Reasoning: Ratios and Rates
- UNIT 6: Expressions, Equations, & Inequalities
- UNIT 7: Geometry
- UNIT 8: Geometry
- UNIT 9: Statistics
- UNIT 10: Review Unit

- LESSON 1: Pre Test
- LESSON 2: Where Does That Fall On The Number Line?
- LESSON 3: How Close Is "Close"?
- LESSON 4: What Are Integers?
- LESSON 5: Adding and Subtracting Integers on a Number Line
- LESSON 6: Adding and Subtracting Integers with Counters
- LESSON 7: Rational Numbers and Integer Practice
- LESSON 8: Show What You Know About Integers and Rational Numbers
- LESSON 9: Absolute Value and Stocks
- LESSON 10: Tracking Stocks and the Coordinate Plane
- LESSON 11: Tracking Stocks and Distance on the Coordinate Plane
- LESSON 12: Tracking Investments and Review
- LESSON 13: Show What You Know + Stocks Wrap Up
- LESSON 14: What Rides Can You Go On?
- LESSON 15: Inequalities
- LESSON 16: Unit Review Stations
- LESSON 17: Unit Test