# Comparing Decimals

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## Objective

SWBAT compare two decimals to hundredths by reasoning about their size and using a number line model.

#### Big Idea

Students will categorize decimals based upon their relative sizes and then they will place the decimal cards on a number line.

## Opening

20 minutes

Today's Number Talk

For a detailed description of the Number Talk procedure, please refer to the Number Talk Explanation. For this Number Talk, I am encouraging students to represent their thinking using a number line model and hundreds grids. For each task today, students shared their strategies with peers (sometimes within their group, sometimes with someone across the room). It was great to see students inspiring others to try new methods and it was equally as great to see students examining each other work for possible mistakes!

Getting Started

Prior to the lesson, I placed magnetic money and fractions on the board to help students conceptualize our number talk today.

I invited students to get a Student Number Line and Hundred Grids. I then drew a Number Line on the Board and marked 0, 1, and 2 on the line. I asked students to do the same on their own number lines.

Task #1: 1  2/10 - 90/100

To begin, I asked students to subtract 2/10 - 90/100 on their number lines and hundreds grids. During this time, some students chose to work alone while others worked with a partner in their math groups. I took this time to conference with students. Here are a few examples of student work during this time: 1 2:10 - 90:100 Student Hundreds Grid and 1 2:10 - 90:100 Student Number Line.

Students then volunteered to model their thinking on the board. During this time, I continually asked clarifying questions to encourage students to make connections between fractions and decimals and to consider more exact language.

Task #2: 2  3/10 - 7/10

Next, we moved on to adding 2 3/10 - 7/10. Once again, students showed their thinking using a number line, 2 3:10 - 7:10 Student Number Line, and hundreds grids, 2 3:10 - 7:10 Hundreds Grid.

Although I didn't capture it on film, students then modeled their thinking on the board.

## Exploration

70 minutes

Goal & Lesson Introduction

I began today by introducing the goal: I can compare two decimals to hundredths by reasoning about their size and using a number line modelI explained: Today, you will be given decimal cards and you will work together with your partner to compare the decimals.

Choosing Partners

Picking math partners is always easy as I already have students placed in desk groups based upon behavior, abilities, and communication skills. Before students began working, I asked them to discuss how they would like to support each other today. I gave them many examples: Do you want to take turns talking out loud? Do you want to solve quietly and then check with each other? Or do you want to turn and talk anytime you get stuck? Students always love being able to develop a "game plan" with their partners!

Categorizing Decimal Cards

I passed out copies of these Decimal Comparison Cards to each set of partners. I asked students to begin cutting out the cards and to determine the best way of categorizing the cards.

At first students wanted more direction, but with time and conversation with their partners, they decided to use the cards in the top row (Near 0, About 1/2, Close to 1) as headings for each of their categories. Next, they used markers to help distinguish each category. For example, one group colored all the cards in the "Close to 1" category blue.

By including the category labels (Near 0, About 1/2, Close to 1), students reasoned about the size of each decimal number to determine the correct category.

Also, as students finished, they labeled the bottom row of blank cards with their own decimal numbers. Students really enjoyed coming up with their own cards!

During this time, I continually encouraged students to use correct language when discussing each decimal number. Instead of saying "0 point 56" to identify 0.56, I encouraged students to say, "Fifty-six hundredths."

Here's an example of the end product: Categorizing Decimal Cards.

Constructing Decimal Number Lines

Following this activity, I passed out construction paper and a meter stick to each pair of students. I explained the next step: Now, I'd like to challenge you to create a number line using your construction paper to show how the decimal numbers on your cards relate to one another.

Again, I gave minimal instructions in order to engage students in Math Practice 1: Make sense of problems and persevere in solving them. I wanted students to "analyze givens, constraints, relationships, and goals" in order to figure out how to use the tools provided to represent their thinking.

At the end of this lesson, I wanted revised student posters (with mistakes crossed off and corrected). Having the opportunity to make mistakes and find solutions is essential to developing problem solving skills.

Monitoring Student Understanding

While students were working, I conferenced with every group. My goal was to support students by providing them with the opportunity to explain their thinking and by asking guiding questions. I also wanted to encourage students to construct viable arguments by using evidence to support their thinking (Math Practice 3).

1. Can you explain what you know?
2. What step did you take first?
3. Does that make sense to your partner too?
4. What are you thinking?
5. Can you show me your thinking?
6. What fraction is this decimal equivalent to?
7. How do you say this decimal number?

Conferences

During this conference, Placing Decimals on a Number Line, students worked together to find the correct placement of each decimal card on the number line. Here's this group's end product: Pink Student Number Line.

Here, Using Benchmark Numbers, a group explains how to use the halfway point between 0 and 0.50 (0.25) to determine if a decimal is closer to 0 or a half.

Closure

To bring closure to this activity, I asked students to place their number lines on the board and join me on the front carpet: Displayed Student Work on the Board. Here are a few examples of student number lines: Examples of Student Number Lines

We then discussed the student number lines as a group. Here, one student points out the importance of attending to increments between decimal numbers on the number line: Decimal Comparison Cards

## Independent Practice

20 minutes

Practice Page

Whenever students have the opportunity to explore a math concept beyond a pencil and paper, I always try to provide some paper practice thereafter to ensure that student are able to transfer their learning to other applications.

For the last 20 minutes of today's lesson, I provided students with two decimal comparison pages (front to back) found at CommonCoreSheets. Surprisingly, students zipped through these two pages quickly!

Here, Relating Decimals to Money, I relate decimals to money to help a student make further sense of decimals comparisons.