# Examining Decimal Patterns

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## Objective

SWBAT analyze patterns with decimal numbers.

#### Big Idea

Using a decimal grid, students will identify, compare, and analyze decimal patterns.

## 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: 1/10 + 0.25

To begin, I asked students to add 1/10 + 0.25 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.

Next, some students volunteered to come up to the board to show their thinking: Student Modeling 1:10 + 0.25.

Others watched carefully, checking their own number lines and hundreds grids to make sure they agreed with the students demonstrating their thinking. Here are a few examples of student work during this time:

Task #2: 0.56 + 6/10

Next, we moved on to adding 0.56 + 6/10. Again, students solved this problem using the number line, 0.56 + 6:10 Number Line, and hundreds grids, 0.56 + 0.6 Hundreds Grids

A few students excitedly volunteered to explain their thinking on the board: Students Modeling 0.56 + 6:10

Some students constructed some pretty impressive number lines: Impressive Number Line.

## Teacher Demonstration

40 minutes

Goal & Lesson Introduction

To begin today's lesson, I shared today's goal: I can analyze patterns with decimal numbers. I explained: To truly understand decimals, it's important to take the time to analyze decimals further. I can't wait to see what you observe today!

Encouraging students to look for patterns is an important part of engaging students in Math Practice 7: Look for and make use of structure. By doing this, students begin to see the bigger picture, which helps them gain perspective and develop a deeper understanding of 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.

Decimal Grid

I passed out a Decimal Board, found at Virginia Department of Education, to each student and asked math partners to begin documenting observations off to the side of the decimals grid.

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. What pattern did you notice?
4. Does that make sense to your partner too?
5. Can you show me your thinking?
6. What happens as you move down a column?
7. What does this chart count by?

Conferences

At first, many students thought this was similar to a multiplication chart: Looks Like a Multiplication Chart. I contemplated teaching my students how to multiply decimals as a "side bar lesson" but decided to respond with, "Does that seem right?"

Another group realized that the chart counted by ones. With time, they realized that it really counted by hundredths: Counting by Hundredths

This group, Similar Digits, noticed that there were common digits in the hundredths place down each column (0.02, 0.12, 0.22, 0.32): Similar Digits.

I liked how this group differentiated between one tenth and one hundredth: One Hundredth vs One Tenth

Finally, this student, One Tenth = Ten Hundredths, noticed that you can just add a "0" on to 0.1 because 0.1 = 0.10.

At this point, we discussed some of the observations that students noted on the sides of their paper (Student Observations A and Student Observations B) as a class.

Making a Fraction Board

Using a Blank Hundreds Chart, I asked students to construct a Fraction Board similar to the Decimal Board. This took some time, but it was a powerful opportunity for students to truly see the relationship between decimals and fractions. Here's a completed Fraction Board.

Afterwards, we discussed how the Decimal Board and Fraction Board were similar. Eventually, the class decided, "Decimals are the same thing as fractions! Fifty Hundredths equals 50/100 and 0.50!"

## Student Practice

40 minutes

Continued Practice

To provide students with more practice comparing and analyzing decimals using their decimal boards, I retyped three sets of questions: Decimal Questions, found at Virginia Department of Education. After making a copy of questions for each group, I cut them into sets so that I could provide groups with one "challenge" at a time.

Challenge 1: Comparing Decimals

I passed out the first sheet of questions, starting with the question, "Which is less 0.01 or 0.02?" We discussed the solution to this question as a class. Then, students continued on their own: Which is More?.

Challenge 2: Subtracting Decimals

As students finished the first challenge, I passed out the next set of questions and we discussed the first question, "What is two hundredths less than thirty-five hundredths?" Again, we discussed the answer to the first question prior to students moving on to the next question in the set: Subtracting Hundredths.

Challenge 3: Completing Number Patterns

Finally, I passed out the last set of questions, with incomplete number patterns, "0.05, 0.01, ___." Once again, we discussed the answer to the first problem and then students were excited to continue on with their partners: Completing Patterns.