Adding and Subtracting Decimals
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1 Adding and Subtracting Decimals LAUNCH (8 MIN) Before When is the sum of two decimals a decimal? When is the sum a whole number? Explain. Your friend says all decimals are less than 1. Do you agree? Explain. During What operations do you use to find your total cost and then find your change? After What is the difference between finding the total cost of the two clowns and finding the total cost of the two clowns plus the two deluxe noses? Use place value names in your answer. PART 1 (6 MIN) Do you line up the decimal points when adding whole numbers? Why? Do you need to write the extra zero for 15.09? Before solving the problem Why would you care about the length of the suspended portion of the bridge? Kala Says (Screen 2) Use the Kala Says button to describe how the main span is used to compare suspension bridges. Do you use different rules for adding place values to the left side of the decimal point from the rules for adding on the right side? PART 2 (6 MIN) How is this algorithm similar to the algorithm for adding decimals? How are they different? Kala Says (Screen 2) Use the Kala Says button to describe how runners use a different method when rounding 100-meter dash times. How can you use the times in this problem to estimate the time for another runner in the 1968 Olympics? PART 3 (6 MIN) Kala Says (Screen 1) Use the Kala Says button to connect adding and subtracting decimals to adding and subtracting whole numbers. What are two steps that may be necessary to vertically line up like place values for addition or subtraction? How can you check a sum or a difference without using a calculator? KEY CONCEPT (6 MIN) Students review the key idea from the lesson: how to set up an addition or subtraction problem, starting with lining up the decimal points and writing zeros as placeholders, if necessary. CLOSE AND CHECK (8 MIN) Why aren't whole numbers enough? Why do you need decimals? Your friend claims that adding decimals is similar to adding whole numbers because in both cases, you line up the decimal points. How can your friend be correct if there are no decimal points in whole numbers?
2 Adding and Subtracting Decimals LESSON OBJECTIVES 1. Add multi-digit decimals. 2. Subtract multi-digit decimals. FOCUS QUESTION How can you add and subtract decimals as easily as you do whole numbers? MATH BACKGROUND While students have worked with decimals in previous grades, this topic refreshes students on operating with decimals and converting between fractions and decimals in preparation for their work in the topic Rational Numbers. Students will recall prior skills of reading, writing, and comparing decimals up to the thousandths, and adding and subtracting decimals up to the hundredths. While fractions and decimals are interrelated, students will not explicitly make this connection until later in this topic. In the first lesson, students recall how to add and subtract decimals up to the thousandths, including decimals with different numbers of decimal places. One of the common ideas in this topic is that of an algorithm. Each operation involving decimals is taught as a sequence of steps. The algorithms for addition and subtraction are combined into the same lesson because they are so similar in nature. Students should use what they know about addition and subtraction as related operations. As students progress through this topic they will use and compare specific algorithms, explore the rules of algorithms, and begin to discuss the roles that algorithms have in mathematics. LAUNCH (8 MIN) Objective: Solve a real-world problem by adding and subtracting decimals. Students add and subtract with whole numbers, then add and subtract with decimals. They observe that working with decimals extends the number of place values to the right of the ones place and decimal point for adding or subtracting but does not change the step-by-step processes for those operations. Before When is the sum of two decimals a decimal? When is the sum a whole number? Is the difference of a whole number and a decimal always a decimal? Explain. [Sample answer: The sum of two decimals is usually a decimal, but the sum can be a whole number if the sum of the decimal parts is 1. The difference of a whole number and a decimal number is always a decimal number.] Your friend says all decimals are less than 1. Do you agree? Explain. [I disagree; numbers greater than 1, such as 1.28 and 12.4, are also decimals.] During What operations do you use to find your total cost and then find your change? [Sample answer: You add to get the total cost. You subtract to find the amount of change.] After What is the difference between finding the total cost of the two clowns and finding the total cost of the two clowns plus the two deluxe noses? Use place value names in your answer. [Sample answer: Finding the total cost of the two
3 clowns involves addition using the ones and tens places. Finding the total cost for the clowns and noses involves using the hundredths, tenths, ones, and tens places.] Solution Notes This activity prepares students to perform addition and subtraction with decimals. If students show specific addition or subtraction examples as part of their explanations, be sure they vertically align the decimal points. Connect Your Learning Move to the Connect Your Learning screen. Start a conversation about how when you're talking about money, decimals mean coins. Listen for students to describe the number after the decimal point as part of a dollar. PART 1 (6 MIN) Objective: Solve word problems by adding multi-digit decimals. Students learn an algorithm for adding decimals. They solve a problem by adding two decimals, one with two decimal places and the other with three. Students will repeat this process with subtraction in Part 2. Instructional Design Have students click on the radio buttons to launch animations that describe the steps for adding two decimal numbers. Emphasize lining up the decimal points and writing in zeros if one number has more decimal places than the other. Move to Screen 2. Introduce the real-world problem and help students understand that they can solve it using addition. Do you line up the decimal points when adding whole numbers? Why? [Yes; the decimal point may not be written but is on the very right of any whole number. You can only add like place values.] Do you need to write the extra zero for 15.09? [Sample answer: No; it is more important to line up the decimal points before adding.] Before solving the problem Why would you care about the length of the suspended portion of the bridge? [Sample answers: Tourists may be interested, especially because at one time it was the longest suspension bridge in the world. Engineers would be interested to know how much support a bridge needs based on the length of its suspended portion.] Kala Says (Screen 2) Use the Kala Says button to describe how the main span is used to compare suspension bridges. Discuss how it is important to use common language so that comparisons are fair. How is the suspended portion related to the main span? How is the total length related to the suspended portion? [The suspended portion is the main span plus the side spans, which are the parts of the bridge that are suspended outside the main span. The total length is the suspended portion plus the parts of the bridge that are not suspended.]
4 Do you use different rules for adding place values to the left side of the decimal point from the rules for adding on the right side? [Sample answer: The only difference is that you may have to add zeros to the right of the decimal point if the two addends have different numbers of decimal places. After that, the process is the same.] Solution Notes Emphasize how adding decimals is remarkably similar to adding whole numbers. Students should pay attention to the additional steps in the algorithm but add as they normally would otherwise. Students may wonder why they need to write zeros to the right of the decimal point when they do not have to write zeros to the left of the decimal point. Explain that it is easier to misalign the digits to the right of the decimal point. Error Prevention Remind students that the purpose of adding zeros to the right of a decimal point is to avoid the error of adding digits from different place values, such as tenths and hundredths. Have students line up the decimal points before they begin adding. Got It Notes While the Example includes the step of writing a zero in the thousandths place of one of the addends, in this problem both addends have the same number of decimal places, so students can just line up the decimal points and add. If you show answer choices, consider the following possible student errors: Students who choose A are probably subtracting. If students did not regroup placevalue amounts correctly, they may choose B or C. Got It 2 Notes This question asks students to consider why you are allowed to add zeros to the end of a decimal. Students should understand that the digits have no value and are merely acting as a placeholder. Compare it to writing zeros at the left of a number, such as , which does not change its value. PART 2 (6 MIN) Objective: Solve word problems by subtracting multi-digit decimals. Now students learn an algorithm to subtract decimals in order to solve a real-world problem. They consider the similarities between the two algorithms in this lesson. Instructional Design Use the radio buttons in the Intro to step students through the algorithm for subtracting two decimals. Note that the first two steps are identical to those for adding decimals. On Screen 2, students can repeat the steps by solving the problem on the whiteboard. How is this algorithm similar to the algorithm for adding decimals? [The first two steps of lining up the decimal points and adding an extra zero to make sure the numbers have the same number of decimal places are identical.]
5 How are the algorithms different? [Sample answer: You subtract each column of digits instead of adding. You regroup place-value amounts differently when subtracting.] Kala Says (Screen 2) Use the Kala Says button to describe how runners use a different method when rounding 100-meter dash times. Each time is recorded to the thousandth, but then rounded up to the next highest hundredth of a second. What are some times to the thousandth place that round to 9.95 seconds? What are the least and greatest values that do? [Sample answer: Two thousandthplace values that round to 9.95 seconds are seconds and seconds. The least is seconds, and the greatest is seconds.] How can you use the times in this problem to estimate the time for another runner in the 1968 Olympics? [Sample answer: The time is probably between 9.95 seconds and 14.7 seconds but much closer to 9.95 seconds.] Differentiated Instruction For struggling students: Students may need a reminder about the meaning of regrouping place-value amounts in terms of subtraction. You can review subtraction with whole numbers as needed to make sure students are prepared to subtract decimals after completing the first two steps of the algorithm. You should particularly discuss what to do if the bottom digit is greater than the top digit. For advanced students: Ask students to find the distance each time (9.95 and 14.7) is from 10 seconds, and what those distances might tell you about running 100 meters in 10 seconds. Got It Notes This question supplies two extra data values, so students' first step is to identify which particular values to use. If you show answer choices, consider the following possible student errors: Students who choose B or C probably calculated the difference between the wrong tubes. If students used the correct tubes but placed the decimal point incorrectly, they may select D. PART 3 (6 MIN) Objective: Add and subtract multi-digit decimals. Students practice adding and subtracting decimals, observing that both algorithms have similar steps that are unique to these two operations. Students should focus on the two preliminary steps that result in the same procedure as the one they already know for adding and subtracting whole numbers. Kala Says (Screen 1) Use the Kala Says button to connect adding and subtracting decimals to adding and subtracting whole numbers. What are two steps that may be necessary to vertically line up like place values for addition or subtraction? [Sample answer: By vertically aligning the decimal points and including zeros as placeholders, each column of digits will have two digits with the same place value.]
6 How can you check a sum or a difference without using a calculator? [Sample answer: For addition, you can subtract one of the addends from the sum and check that you get the other addend. For subtraction, you can add the difference to the subtrahend and make sure you get the minuend.] Solution Notes Finding sums and differences in the same problem allows you to focus on the similarities in the algorithms. Use the provided solution to demonstrate the three steps using color. Students may point out that the regrouping of place-value amounts is the one place where the two processes differ. Error Prevention If students are making errors lining up the decimal point and lining up columns of like place values, encourage them to use grid paper, with one column for decimal points and one digit in each cell. Lined paper, turned vertically, also provides a helpful guide for recording place values correctly. Got It Notes While the Example included both addition and subtraction problems, this Got It presents a single subtraction problem. If you show answer choices, consider the following possible student errors: Students who left out a digit may choose A or B. If they added the two decimals, they probably will select D. KEY CONCEPT (6 MIN) Teaching Tips for the Key Concept Students review the key idea from the lesson: how to set up an addition or subtraction problem, starting with lining up the decimal points and writing zeros as placeholders, if necessary. Stress that the processes of addition and subtraction are similar to each other for all types of numbers, including whole numbers and decimals. Why do you need to line up the decimal points? [Sample answer: You can only subtract pairs of digits that have the same place value.] CLOSE AND CHECK (8 MIN) Focus Question Sample Answer You can use place value to add and subtract decimals just like you do for whole numbers. The only differences are that you may need to write zeros as placeholders and you need to remember to write the decimal point in the answer. Focus Question Notes Look for students to describe how adding and subtracting whole numbers is similar to adding and subtracting decimals: in both cases, you line up the decimal points of each number. But when you add whole numbers, you don't have to write the decimal point. Essential Question Connection This topic addresses the portion of the Essential Question that asks how you can extend your knowledge of place value to operations with decimals. Here, students
7 learn that when adding or subtracting decimals, you need to line up the decimal points to ensure that all place values are lined up as well. Why aren't whole numbers enough? Why do you need decimals? [Sample answer: Decimals are more exact than whole numbers. Sometimes a measurement is not exactly 5 or 6 units, but falls between 5 and 6 units. Decimals are a way to communicate this "in between" number.] Your friend claims that adding decimals is similar to adding whole numbers because in both cases, you line up the decimal points. How can your friend be correct if there are no decimal points in whole numbers? [Sample answer: Your friend knows that whole numbers have a decimal point after the rightmost digit (after the ones place value), but you don't have to write it.]
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