• Evaluation of presentations during partner activities (Compare Pairs and Uptown or Downtown) will help in determining student level of understanding.
• Observation during Decimal Match group activity will aid in assessing comprehension level of class.
• The Lesson 3 Quick Quiz may be used to gauge student mastery of lesson concepts.

“In the previous lessons you practiced writing multiple forms of decimal numbers, evaluated meanings of the different place values, multiplied by powers of ten, and divided by powers of ten. In this lesson we will work on two more skills used with decimals. The work you have already done will help you with this. In this lesson you will practice comparing and rounding decimals.”

Write the following or similar comparisons on the board:

“Which is larger?”   

“Take a minute to compare the fractions on the board.

“What are some easy ways to compare?”  (Draw pictures of them, double the denominator of one, get a common denominator, etc.)

“Are they all easy to compare?” (no) “Why not?” (The denominators 17 and 33 are hard to compare.)

“What strategy might help us compare any set of fractions with unequal denominators?” (Finding common denominators.)

“Decimal numbers are another way to represent fractional parts of a whole. When we compare decimals we need to use a strategy that is similar to finding a common denominator, but instead of basing our comparison on denominators, we base them on the powers of 10, which create our decimal place values.”

“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?” Call on one or more students to explain. If students do not bring up the comparison of  and  do so yourself, or ask guiding questions to get the students to discover this method for comparing.

“This time I am going to offer you 0.4 parts or 0.34 parts of the 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 0.34 is the larger piece?” Again allow students to respond. Listen for and correct the common misconception that can occur here. Some students will see that 34 looks larger than 4 so they may assume 0.34 is more pizza. Be sure students use the denominators, pictures, or another method to show that , while 0.34 =  (so 0.34 is actually smaller).

“With a partner, decide which is larger 0.512 or 0.52 of the pizza. Be prepared to explain why you think so.” Give students 1–3 minutes, and then ask students to share. Although many ideas may be shared, be sure students see the connections of 0.512 =  while 0.52 = , which can be rewritten as the equivalent .  In a pizza cut into a 1,000 tiny pieces, 520 would be slightly larger than 512 of the same size pieces.

“Discuss with your partner a strategy that you think will help us compare decimals like these easily.”

Ask students to share their strategies. Emphasize all correct reasoning. Be sure to bring up the following strategies if students do not:

• Compare one digit at a time starting in the furthest left place value (largest place value). If the digit is the same, move right one place to compare the next digit. Continue until the place value digits are different. The number with the larger digit in this place is the larger overall value. If a digit is missing, treat it as a 0.
• Even out the length of the decimal numbers by placing one or more nonsignificant zero(s) at the end of the shorter decimal. By doing this, the place values will become the same. This is an easy method of finding a common denominator. Both decimals will have the same place value, such as hundredths or thousandths, which makes a direct comparison possible.

“Consider which is larger, 0.50 or 0.5? Can we use our strategies to help us decide?” (yes)

“If we compare one digit at a time, both have zero ones, and both have five in the tenths place. So far they have the same values. In the hundredths place 0.50 has 0 parts shown.  Nothing is shown in the hundredths place for 0.5, which is the same as none or zero. Since each place value in the two numbers is equivalent, the numbers themselves are equivalent.”

“You could also have used the strategy of placing one or more nonsignificant zeros, as needed, to create the same place value for both decimal numbers. In this case, 0.50 would be compared to 0.50 after placing a zero at the end of 0.5. Compare 0.50 to 0.50 or  to . They are equal.”

Do several more comparisons with the class, demonstrating both methods. Remind students of the symbols used to compare (<, >, or =). Use these or similar examples:

• 7.54   ?  7.539             (>)
• 0.61   ?   0.610            (=)
• 6.009  ?   6.09             (<)
• 3.010  ?   2.99             (>)
• 0.0712  ?  0.701          (<)

Partner Activity: Compare Pairs

Give students 5–10 minutes to work with a partner to complete the Compare Pairs sheet (M-5-5-3_Compare Pairs and KEY.docx). Monitor students while they work. Give further assistance to students who are struggling. When the sheets are complete, ask each pair of students to explain one problem and the process they used to solve it. Have other students ask questions or identify and correct any mistakes they hear their classmates make. Do additional examples as necessary before moving on to the next activity.       

Begin the discussion on rounding decimals by reviewing the process of rounding whole numbers.

“Recall that to round a whole number to any specific place value, we use the digit to the _________ of the digit you are rounding to determine how to round. If the digit is ________ or greater, round up one. If it is _______ or less, round off (never round down). Use place-holding zeros to fill in for every digit that is rounded off.” (right, five, four)

“For example, let’s round 586 to the nearest ten. Do this on your paper.

“I want you to underline the digit in the tens place, 586 since we are rounding to the nearest ten. Now, draw a vertical separation line to the right of the tens place, 58│6.

“Next, circle the digit just to the right of the separation line,                                                        

“The circled digit is greater than five, so we will round the underlined digit up one. Every digit after the separation line gets rounded off and replaced with a place holding zero. Our answer is _____.” (590)

Repeat these steps for 9,139 rounded to the nearest hundred. State the steps one at a time as students work. They should get 9,100 when they are done with these steps.

• Underline the digit in the hundreds place.                             (1)
• Draw a vertical separation line after the underlined digit.     (between 1 and 3)
• Circle the first digit to the right of the separation line.          (3)
• Use the circled digit to decide if you round up or off.          (round off)
• Use place-holding zeros to fill rounded off digits.                 (in place of the 3 and 9)

Review additional whole number rounding examples if students need them before discussing rounding decimal numbers.

“Rounding decimal numbers is an almost identical process to what we just reviewed. The only difference is that we do not use the place-holding zeros to fill rounded-off decimal digits. Let’s try a few together.”

• 3.7428 to the nearest one                    (4)
• 3.7428 to the nearest tenth                 (3.7)
• 3.7428 to the nearest hundredth         (3.74)
• 3.7428 to the nearest thousandth        (3.743)

Talk students through rounding to the nearest one and nearest tenth. Then have them do the others on their own. For rounding 3.7428 to the nearest one (or whole):

• Underline the digit in the ones place.                                     (3)
• Draw a vertical separation line after the underlined digit.     (between 3 and 7)
• Circle the first digit to the right of the separation line.          (7)
• Use the circled digit to decide if you round up or off.          (round up)
• Do not use place-holding zeros to fill rounded-off digits if they are to the right of the decimal point.             (None are needed here, so the answer is 4.)

For rounding 3.7428 to the nearest tenth:

• Underline the digit in the tenths place.                                   (7)
• Draw a vertical separation line after the underlined digit.     (between 7 and 4)
• Circle the first digit to the right of the separation line.          (4)
• Use circled digit to decide if you round up or off.                (round off)
• Place-holding zeros are not needed here because the rounded-off digits are to the right of the decimal point.                                                                         (3.7)

When students have finished the last two problems, select a student to demonstrate the process for each on the board. (hundredths: 3.74, thousandths: 3.743)

Partner Activity: Uptown or Downtown

Give students approximately 10–15 minutes to complete the Uptown or Downtown activity sheet together (M-5-5-3_Uptown or Downtown and KEY.docx). Circulate around the room asking questions to determine each student’s level of understanding. Assist students who need more guidance. When the sheets are complete, ask each pair of students for an explanation for one problem and the process they used to round it. Have other students ask questions or identify and correct any mistakes they hear their classmates make. Do additional examples as necessary before moving on to the group activity.

Group Activity: Decimal Match Game

Place students in groups of two to four for this game. Each group will need one set of 30 game cards (M-5-5-3_Decimal Match Cards 1.docx or M-5-5-3_Decimal Match Cards 2.docx). The cards should be in random order and placed face down to form a rectangular grid 5 x 6 cards. Students will need paper and pencil to record their work as cards are revealed. Each player will flip over any two cards looking for a match. A match consists of a decimal question and the correct answer to that question. If it is not a match, the cards are returned to the table face down in the same location. If two answer cards or two question cards are selected, there cannot be a match.

Modifications

• You may choose for some groups to place all question cards to the left and all answer cards to the right to make matches a bit easier to find.
• You may also reduce the number of cards for a group by removing some matched pairs of cards.

Remind students to show the problems and work on their paper because this will help them as the game progresses. When a player finds a match, the player keeps it. It must be placed face up in front of them until you can check that it is a correct match. Players who find a match on their turn get to try another pair on the same turn.

As students play, monitor the work they show on their paper and the face-up pairs they have found. Once a pair is checked, and you tell them it is correct, the student will keep the pair, but should flip cards face down so you know it has been checked. If you detect incorrect work on student papers, assist in correcting errors. If you find a student has matched incorrect pairs, discuss the error in logic and return the cards to the playing surface face down so they can be matched correctly with the other cards still in play. Play continues until all pairs have been found. Groups that finish quickly can play again using the second version of the cards.

Have each student complete the Lesson 3 Quick Quiz (M-5-5-3 Lesson 3 Quick Quiz and KEY.docx). Use the results of this assessment and observations from the lesson activities to determine which of the optional instructional strategies below may be used for each student.

Extension:  

• Routine:  During the school year, mention instances where rounding or decimal comparisons are used for practical daily tasks such as shopping, cards, and sports averages. Ask students to discuss examples of rounding or comparing decimals that they notice both in and out of school. Encourage them to cut out and bring examples of stories, tables of data, survey data, or nutrition labels that could be used to practice rounding or comparing in a class activity.
• Small Group: These stations may be used for students who are having difficulty with the concepts of decimal place comparisons or rounding. Students can work alone or in pairs.

      Work Station 1: Ultimate Number Activity

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). On each card in a set of 5 cards, draw a large single digit or the decimal point with a marker. Make additional sets on different colors.

Teacher direction is needed for this station. One to three groups of four or five students can do this activity at the same time. Hand each student in one group a digit card or decimal card from the same color pile. Do the same from a different color pile for each additional group. All decimal cards need to be handed out. Any extra digit cards can be set aside. Ask students to gather in groups with the other students who have the same card color.

“When I give the signal, I would like you to work silently within your group to arrange yourselves in a way that creates the greatest possible decimal value with your digit cards. Remember that no talking is allowed. Are there any questions before we begin? Begin!” Allow students approximately 1–2 minutes to move into position. Point out the answers of each group. If some groups do not have the largest number they could make, use questioning to help them rearrange. Have a quick discussion with questions similar to:

• “How do you know this is the largest value?”
• “Can we trade any digits to make the value larger?”
• “Can you move your decimal point to make the value larger?”
• “What is the most important digit in making your number the greatest value it can be?”

Move on to the next part of the activity.

      “Rearrange yourselves to make the smallest possible decimal number. I will be back in a minute or two to check. Think about the questions I just asked you, and how they can help you check your own number this time.”

      Check student numbers and assist if needed.

      “Can you change the location of any digit or your decimal point to make your value any smaller? What did you need to think about this time?”

Work Station 2: Decimal Match Game

Place the second version of the Decimal Match Game cards at this station (M-5-5-3_Decimal Match Cards 2.docx). Post the directions to remind students how to play. Break the set of cards up into several smaller sets of cards to make the pairs easier to find. Students can do this activity individually or in a small group.

Work Station 3: Compare Decimals with Base-ten Blocks

Place base-ten blocks at this station. Post directions stating what each block represents.

Students will compare decimal values to find the greatest and least by combining base-ten blocks to visualize the size comparisons. Each student at this station will need a record sheet to fill in (M-5-5-3_Compare Decimals with Blocks and KEY.docx).

• Expansion: Beat My Value Game

Use this activity for students who have mastered the content of this lesson. This will give them an opportunity to develop game strategies based on place value, and number value comparisons. Students will play in groups of 2 to 6 players. Each group will need one
10-sided number cube or a 10-section spinner marked with the digits 0–9 (M-5-5-3_Game Spinner 10 Section.docx) and a paper clip to spin with. Each player will need the Beat My Value record sheet (M-5-5-3_Beat My Value Game.docx).

Players will take turns rolling the number cube or spinning the spinner once. When a digit from 0–9 is selected, each player fills in any one space in his/her number for the round. Play for one round continues until the digits for that number are completely filled in. At this time, players compare answers. The goal is to be the player with the greatest number. The record sheet has room for eight rounds. The playing spaces for rounds four and eight are shaded gray. In these two rounds, students are trying to create the smallest number possible. At the end of the game, the student who won the most rounds is the overall game winner (even if there was not time to play all eight rounds).