<p>In this lesson, students will continue to work with decimal numbers. The concept of place value will be utilized. Students will:&nbsp;<br>- compare two decimal numbers, use the symbols &lt;, &gt;, and =.&nbsp;<br>- sort a set of decimal values with varying place value lengths from least to greatest.&nbsp;<br>- write similar decimal numbers with zeros.&nbsp;<br>- round decimal numbers to any place value, from one to thousandths.&nbsp;</p>

<p>- How can mathematics help to quantify, compare, depict, and model numbers?<br>- How can mathematics help us communicate more effectively?<br>- How are relationships represented mathematically?<br>- What makes a tool and/or strategy suitable for a certain task?<br>- How may patterns be used to describe mathematical relationships?</p>

<p>- Decimal Place Value: A place value to the right of the decimal point in a number. The base for a decimal place value is less than 1. Each place value after the decimal point is 1⁄10 of the place value to its left.<br>- Expanded Form: A method for representing a base-ten number as the sum of its parts, each represented by its base value multiplied by a power of ten.<br>- Exponent: A number used to show the number of times a base value should be multiplied repeatedly by itself.<br>- Hundredths Place: The place value two places to the right of the decimal point in a base-ten number. The digit located in this place represents a fractional part out of one hundred.<br>- Tenths Place: The place value on the right of the decimal point in a base-ten number. The digit in this place represents a fractional part out of ten.<br>- Thousandths Place: The place value three places to the right of the decimal point in a base-ten number. The digit in this place represents a fractional part out of one thousand.</p>

<p>- student copies of Vocabulary Journal pages (M-5-5-1_Vocabulary Journal)&nbsp;<br>- student copies of the Compare Pairs sheet (M-5-5-3_Compare Pairs and KEY)&nbsp;<br>- student copies of the Uptown or Downtown activity sheet (M-5-5-3_Uptown or Downtown and KEY)&nbsp;<br>- copies of the Decimal Match Game cards sets (M-5-5-3_Decimal Match Cards 1 and M-5-5-3_Decimal Match Cards 2), print and cut apart one or both sets per small group of students, store each set in a zip-top bag or envelope prior to class&nbsp;<br>- 1 copy of each Decimal Match Game Card set (M-5-5-3_Decimal Match Cards 1 and M-5-5-3_Decimal Match Cards 2), uncut to use for a KEY&nbsp;<br>- student copies of the Lesson 3 Quick Quiz (M-5-5-3_Lesson 3 Quick Quiz and KEY)&nbsp;<br>- <i>optional:</i> 9" x 12" colored construction paper or 8.5" x 11" colored printer paper, used for the Ultimate Number Activity&nbsp;<br>- <i>optional:</i> Comparing Decimals with Blocks (M-5-5-3_Compare Decimals with Blocks and KEY), base-ten blocks are needed for each student at the station&nbsp;<br>- <i>optional:</i> Beat My Value record sheet (M-5-5-3_Beat My Value Game) for Extension activity&nbsp;<br>- <i>optional:</i> Game Spinner 10 Section (M-5-5-3_Game Spinner 10 Section) for expansion activity</p>

<p>- Evaluation of presentations during partner activities (Compare Pairs and Uptown or Downtown) will aid in measuring student level of understanding.&nbsp;<br>- Observation during the Decimal Match group activity will help assess the class's comprehension level.&nbsp;<br>- The Lesson 3 Quick Quiz can be used to assess students' comprehension of lesson concepts.&nbsp;</p>

<p>Scaffolding, Active Engagement, Modeling, Explicit Instruction, and Formative Assessment&nbsp;<br>W: Students will apply their knowledge of place value and tens to compare two decimal numbers with greater than, less than, and equal signs. Students will also practice rounding decimal values to several different places.&nbsp;<br>H: Engage students in the class with a pizza-sharing scenario. Students will learn to compare decimals and simple fractions by converting them to decimals.&nbsp;<br>E: Students compare decimal values using various strategies, including as drawing a model, doubling the denominator, comparing digits in the same position, etc. The partner practice activity will allow for more practice and examination of the procedure.&nbsp;<br>R: The Uptown or Downtown activity and the Decimal Match Game will help students review decimal place value, comparison, and rounding concepts. Keep an eye out for possibilities to reteach or clarify.&nbsp;<br>E: The Decimal Match Game and Quick Quiz can be used to assess students' grasp of concepts.&nbsp;<br>T: Use the Extension section to personalize the lesson to your students' specific requirements. Small Group activities are intended for students who may benefit from more practice with decimal concepts, whereas Expansion recommendations are meant for students who have mastered the concepts covered in the lesson but are searching for a challenge beyond the standard. Routine ideas can be utilized throughout the year to revisit concepts from the lesson.&nbsp;<br>O: This lesson builds on prior understanding of comparing and rounding whole numbers. Students adapt the processes of rounding to work with decimal values. The lesson uses the concept of place value to compare and round decimal values. The lesson ends with a group review and a brief individual assessment.&nbsp;</p>

<p><strong>"In previous lessons, you practiced writing several forms of decimal numbers, determining the meanings of different place values, multiplying and dividing by powers of ten. In this lesson, we will concentrate on two more abilities utilized with decimals. The work you've already done will help you with this. In this lesson, you will compare and round decimals."</strong>&nbsp;<br><br>Post the following or similar comparisons on the board:<br><br><strong>"Which is larger?"</strong><br><br>\(3 \over 8\) or \(1 \over 2\) (<i>\(1 \over 2\)</i>)<br><br>\(5 \over 6\) or \(7 \over 12\) (<i>\(5 \over 6\)</i>)<br><br>\(13 \over 17\) or \(20 \over 33\) (<i>\(13 \over 17\)</i>)<br><br><strong>"Spend a minute comparing the fractions on the board.</strong><br><br><strong>"What are some easy ways to compare?"</strong> (<i>Draw pictures of them, double the denominator of one, find a common denominator, and so on.</i>)<br><br><strong>"Are they all easy to compare?"</strong> (<i>no</i>) <strong>"Why not?"</strong> (<i>The denominators 17 and 33 are difficult to compare.</i>)<br><br><strong>"What strategy might help us compare any set of fractions with unequal denominators?"</strong> (<i>Identifying common denominators.</i>)<br><br><strong>"Decimal numbers are another method of representing fractional parts of a whole. When comparing decimals, we utilize an approach similar to finding a common denominator, but instead of denominators, we base our comparisons on powers of 10, which create our decimal place values."</strong><br><br><strong>"If I share some pizza with you, would you rather have 0.4 or 0.3 parts of the pizza, if you are really hungry?"</strong> Call upon one or more students to explain. If students do not bring up the comparison of \(4 \over 10\) and \(3 \over 10\) do it yourself or provide guided questions to encourage students to discover this method of comparison.&nbsp;<br><br><strong>"This time, I'm going to offer you 0.4 or 0.34 slices of pizza. Which should you choose if you are really hungry? By a show of hands, who thinks 0.4 is the bigger piece? Who thinks that 0.34 is the bigger piece?"</strong> Again, enable students to answer. Listen for and correct the common misconceptions that can arise here. Some students will notice that 34 is larger than 4, thus they may conclude that 0.34 is more pizza. Make sure students use the denominators, pictures, or another approach to demonstrate that 0.4 = \(4 \over 10\) = \(40 \over 100\), while 0.34 = \(34 \over 100\) (so 0.34 is actually smaller.)&nbsp;<br><br><strong>"Decide with a partner which portion of the pizza is greater, 0.512 or 0.52. Be prepared to explain why you think so."</strong> Give students 1-3 minutes, then ask them to share. Although many ideas may be similar, make sure students understand the relationship between 0.512 = \(512 \over 1,000\) and 0.52 = \(52 \over 100\), which may be rewritten as the equivalent \(520 \over 1,000\). In a pizza cut into 1,000 little pieces, 520 is somewhat larger than 512 of the same size.<br><br><strong>"Discuss with your partner a strategy that you think will help us compare decimals like these easily."</strong>&nbsp;<br><br>Ask students to share their strategies. Emphasize all correct reasoning. Be sure to discuss the following strategies if students do not:&nbsp;<br><br>Compare one digit at a time, beginning with the furthest left place value (largest place value). If the digit is the same, shift one position to the right to compare the next digit. Continue until the place-value digits are different. The number with the larger digit in this position has the greater overall value. If a digit is missing, treat it as 0.<br><br>To even out the length of the decimal numbers, add one or more nonsignificant zero(s) at the end of the shorter decimal. By doing so, the place values will be the same. This is a simple way for determining a common denominator. Both decimals will have the same place value (for example, hundredths or thousandths), allowing for a direct comparison.&nbsp;<br><br><strong>"Which is larger: 0.50 or 0.5? Can we use our strategies to help us make a decision?"</strong> (<i>yes</i>)<br><br><strong>"When we compare one digit at a time, both have zero ones and a five in the tenths place. So far, they share the same values. In the hundredths place, 0.50 has 0 parts shown. Nothing is shown in the hundredth place for 0.5, which is equivalent to none or zero. Because each place value in the two numbers is equal, the numbers are also equal."</strong>&nbsp;<br><br><strong>"You could also have used the strategy of inserting one or more nonsignificant zeros as needed to create the same place value for both decimal numbers." In this case, 0.50 is compared to 0.50 after adding a zero to the end of 0.5. Compare 0.50 to 0.50, or \(50 \over 100\) to \(50 \over 100\). They are equal."</strong><br><br>Perform more comparisons with the class, demonstrating both ways. Remind students of the symbols used for comparison (&lt;, &gt;, or =). Use these or similar examples.<br><br>7.54 ? 7.539 &nbsp; &nbsp; &nbsp;(&gt;)<br>0.61 ? 0.610 &nbsp; &nbsp; &nbsp;(=)<br>6.009 ? 6.09 &nbsp; &nbsp; &nbsp;(&lt;)<br>3.010 ? 2.99 &nbsp; &nbsp; &nbsp;(&gt;)<br>0.0712 ? 0.701 &nbsp;(&lt;)<br><br><strong>Partner Activity: Compare Pairs</strong><br><br>Allow students 5-10 minutes to work with a partner to complete the Compare Pairs sheet (M-5-5-3_Compare Pairs and KEY). Students should be monitored while working. Provide additional assistance to students who are struggling. When the sheets are finished, ask each pair of students to describe one problem and the steps they took to solve it. Allow other students to ask questions or identify and fix any mistakes they hear their classmates make. Do extra examples as needed before going on to the next activity.<br>Begin the discussion about rounding decimals by going over the technique of rounding whole numbers.&nbsp;<br><br><strong>"Remember that when rounding a whole number to a certain place value, we use the digit to the _______ of the digit being rounded to determine how to round. If the digit is ________ or greater, round up one. If the digit is _______ or less, round off</strong> (never round down)<strong>. Use place-holding zeros to replace any rounded-off digits."</strong> (<i>right, five, four</i>)&nbsp;<br><br><strong>"For example, let's round 586 to the nearest ten. Try this on your paper.&nbsp;</strong><br><br><strong>"I want you to underline the digit in the tens place, 586, because we are rounding to the nearest ten. Now, draw a vertical dividing line to the right of the ten place, 58 | 6.</strong><br><br><strong>"Now, circle the digit just to the right of the separation line,</strong> <img class="image_resized" style="aspect-ratio:50/31;width:7.14%;" src="https://storage.googleapis.com/worksheetzone/images/image.png" width="50" height="31"><br><br><strong>"Since the circled digit is greater than five, we shall round the underlined digit up one. Every digit after the separation line is rounded off and replaced with a place holding zero. Our answer is _____."</strong> (<i>590</i>)&nbsp;<br><br>Repeat these steps for 9,139, rounded to the nearest hundred. As students work, state each step one at a time. They should get 9,100 once they have completed these steps.<br><br>Underline the digit in the hundreds place. (<i>1</i>)&nbsp;<br>Draw a vertical separation line after each underlined digit. (<i>between 1 and 3</i>)<br>Circle the first digit to the right of the separation line. (<i>3</i>)&nbsp;<br>Use the circled digit to determine whether to round up or off. (<i>round off</i>).<br>Use place-holding zeros to fill in rounded off digits. (<i>in place of the 3 and 9</i>)<br>If students require extra whole number rounding examples, review these before moving on to rounding decimal numbers.&nbsp;<br><br><strong>"Rounding decimal numbers is nearly equivalent to the procedure we just discussed. The only difference is that we don't use place-holding zeros to fill in rounded decimal digits. Let's try a few together."</strong><br><br>3.7428 to the nearest one &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(<i>4</i>)<br>3.7428 to the nearest tenth &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(<i>3.7</i>)<br>3.7428 to the nearest hundredth &nbsp; &nbsp; &nbsp;(<i>3.74</i>)<br>3.7428 to the nearest thousandth &nbsp; &nbsp; (<i>3.743</i>)<br><br>Talk to students about rounding to the nearest one and tenth. Then have them complete the rest on their own. How to round 3.7428 to the nearest one (or whole):<br><br>Underline the digit in the one place. (<i>3</i>)&nbsp;<br>Create a vertical separation line after the underlined digit. (<i>between 3 and 7</i>)<br>Circle the first digit to the right of the separation line. (<i>7</i>)<br>Use the circled digit to determine whether to round up or off. (<i>round up</i>)&nbsp;<br>Place-holding zeros should not be used to fill rounded-off digits that are to the right of the decimal point. (<i>None are needed here, so the answer is 4.</i>)<br><br>To round 3.7428 to the nearest tenth:<br><br>Underline the digit in the tenth place. (<i>7</i>)<br>Create a vertical separation line after the underlined digit. (<i>between 7 and 4</i>)<br>Circle the first digit to the right of the separation line. (<i>4</i>)<br>Use the circled digit to select whether to round up or off. (<i>round off</i>)<br>Place-holding zeros are unnecessary here since the rounded-off digits are to the right of the decimal point. (<i>3.7</i>)<br>When students have completed the last two problems, choose one to show the process for each on the board. (<i>hundredths: 3.74; thousandths: 3.743</i>)<br><br><strong>Partner Activity: Uptown or Downtown</strong><br><br>Allow students 10-15 minutes to complete the Uptown or Downtown activity sheets together (M-5-5-3_Uptown or Downtown and KEY). Circulate around the class and ask questions to determine each student's level of comprehension. Assist students that require further guidance. When the sheets are completed, ask each pair of students to explain one problem and the method they used to solve it. Allow other students to ask questions or identify and fix any mistakes they hear their classmates make. Do as many examples as necessary before going on to the group activity.<br><br><strong>Group Activity: Decimal Match Game</strong><br><br>For this game, divide students into groups of two or four. Each group will need a single set of 30 game cards (M-5-5-3_Decimal Match Cards 1 or M-5-5-3_Decimal Match Cards 2). To make a rectangular grid of 5 × 6 cards, arrange the cards in random order and placed face down. Students will need paper and pencil to record their work as the cards are revealed. Each player will flip over any two cards to see if they match. A match consists of a decimal question and the correct answer to that question. If there is no match, the cards are returned to the table, face down and in the same position. If two answer cards or two question cards are chosen, there will be no match.<br><br><strong>Modifications</strong>&nbsp;<br><br>You can arrange some groups so that all question cards are on the left and all answer cards are on the right to make matches easier to find.&nbsp;<br>You can also minimize the number of cards in a group by deleting matched pairs of cards.&nbsp;<br>Remind students to show the problems and work on their paper because it will benefit them as the game proceeds. When a player discovers a match, they keep it. It must be placed face up in front of them until you can ensure that it is the correct match. Players who find a match on their turn are allowed to try another pair on the same turn.<br><br>As students play, keep track of the work they show on their paper and the face-up pairs they discover. Once a pair is checked and you confirm that it is correct, the student should keep the pair but flip the cards face down so you know that it has been checked. If you notice any incorrect work on student papers, please aid in correcting errors. If you discover that a student has matched wrong pairs, examine the error in logic and return the cards to the playing surface face down so that they can be correctly matched with the remaining cards in play. Play continues until all pairs have been discovered. Groups who finish fast can play again with the second set of cards.<br><br>Each student should complete the Lesson 3 Quick Quiz (M-5-5-3 Lesson 3 Quick Quiz and KEY). Using the results of this evaluation and observations from the class activities, determine which of the optional teaching strategies listed below can be used for each student.<br><br><strong>Extension:</strong><br><br><strong>Routine:</strong> During the school year, discuss how rounding or decimal comparisons are used in practical daily tasks like shopping, cards, and sports averages. Ask students to describe examples of rounding or comparing decimals that they see both at and outside of school. Encourage students to cut out and bring examples of stories, data tables, survey data, or nutrition labels that can be used in class activity to practice rounding or comparing.</p><p><strong>Small Group:</strong> These stations may be used for students who are having trouble understanding decimal place comparisons or rounding. Students can work independently or in pairs.&nbsp;</p><p><u>Work Station 1:</u> Ultimate Number Activity&nbsp;<br><br>Make digit cards for this station using colored construction paper or printer paper. Make five cards of one color. Choose four digits and a decimal point (such as 0, 2, 5, 8 and a decimal point). Use a marker to draw a large single digit or the decimal point on each of the five cards in the set. Make additional sets with different colors.<br><br>This station requires teacher direction. This activity can be done at the same time by one to three groups of four or five students. Give each student in one group a digit card or decimal card from the same color pile. Repeat with each additional group, using a different color pile. All decimal cards must be handed out. Any additional digit cards can be set aside. Ask students to gather in groups with other students who have the same card color.&nbsp;<br><br><strong>"When I give the signal, I want you to work silently within your group to arrange your digit cards in a a way that they create the greatest possible decimal value. Remember, no talking is permitted. Do you have any questions before we begin? Begin!"</strong> Allow students around 1-2 minutes to get into position. Point out each group's answers. If some groups do not have the largest number possible, use questioning to help them rearrange. Have a quick discussion with questions such as:&nbsp;<br><br><strong>"How do you know this is the largest value?"&nbsp;</strong><br><strong>"Can we trade any digits to make the value larger?"&nbsp;</strong><br><strong>"Can you move your decimal point to make the value larger?"&nbsp;</strong><br><strong>"What is the most important digit in making your number the greatest value it can be?"</strong><br><br>Move on to the next stage of the activity.<br><br><strong>"Rearrange yourself to make the smallest possible decimal number. I'll come back in a minute or two to check. Consider the questions I just asked you and how they can help you check your own number this time."</strong><br><br>Check student numbers and offer assistance as needed.&nbsp;<br><br><strong>“Can you move the position of any digit or decimal point to make your value smaller? What did you need to think about at this time?”</strong></p><p><br><u>Work Station 2:</u><strong> </strong>Decimal Match Game<strong>&nbsp;</strong><br><br>Place the second version of the Decimal Match Game cards to this station (M-5-5-3_Decimal Match Cards 2). Post instructions to remind students how to play. Divide the set of cards into smaller sets to make it easier to find the pairs. Students can complete this assignment alone or in small groups.</p><p><br><u>Work Station 3:</u> Compare Decimals with Base-ten Blocks&nbsp;<br><br>Set up base-ten blocks at this station. Post directions that explain what each block symbolizes.<br><br><img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_32.png" width="563" height="160"><br><br>Students will compare decimal values to determine the greatest and least by combining base-ten blocks to represent size comparisons. Each student at this station will need a record sheet to complete (M-5-5-3_Compare Decimals with Blocks and KEY).<br><br><strong>Expansion: Beat My Value Game</strong><br>Use this activity with students who have mastered the content of this lesson. This will enable students to create game strategies based on place value and number value comparisons. Students will form groups of 2 to 6 players. Each group will need a 10-sided number cube or a 10-section spinner marked with digits 0-9 (M-5-5-3_Game Spinner 10 Section) and a paper clip for spinning. Each player will need the Beat My Value record sheet (M-5-5-3_Beat My Value Game).<br><br>Players will take turns rolling the number cube and spinning the spinner once. When a digit from 0 to 9 is chosen, each player fills out one spot in his or her number for the round. Play continues for one round until all of the digits for that number have been filled in. At this time, the participants compare answers. The goal is to be the player with the greatest number. The record sheet contains space for eight rounds. The playing areas for rounds four and eight are shaded gray. In these two rounds, students attempt to generate the least possible number. At the end of the game, the student who won the most rounds is the overall game winner (even if there was no time to play all eight rounds).&nbsp;</p>