3.4 DIVIDING DECIMAL NUMBERS

Transcription

1 3.4 DIVIDING DECIMAL NUMBERS Lindy has decided to plant Gerbera daisies along the border of her flower garden. The border is 22.5 feet long. She has been gardening for many years and, based on her experience, she knows that Gerberas require.25 feet of space per plant (so they have the right amount of sunlight and room to grow. She knows what to do with them, once she buys them; she s just not sure how many to buy. Can you help her? How many plants should she purchase? 8 plants Assess your readiness to complete this activity. Rate how well you understand: Not ready Almost ready Bring it on! the long division methodology the positioning of the decimal point in the quotient when and how to use trailing zeros in the dividend when to place zeros in the quotient how to carry out rounding instructions the validation of the answer by multiplication Dividing any two decimal numbers neatness of presentation rounding to the specified place validation of the answer 59

2 Chapter 3 Decimal Numbers While Example is worked out, step by step, you are welcome to complete Example 2 as a running problem. Example : Round the answer to the nearest hundredth. Example 2: Round the answer to the nearest tenth. Try It! Steps in the Methodology Example Example 2 Step Set up the problem. Step 2 Move the decimal point in the divisor. Set up the division problem, carefully recognizing which number is the dividend and which is the divisor. It is not necessary to include leading zero (0 whole numbers in the set-up. Move the decimal point in the divisor to the right of its last digit. For ease of computation in Step 5, it is most efficient to work with a whole number divisor. Step 2 takes care of this, by multiplying the divisor by the appropriate power of ten (Move two places in the divisor. Special Case: The divisor is a whole number (see Model 2 & Model 5 Step 3 Move the decimal point in the dividend. Move the decimal point in the dividend the same number of places to the right as you did in the divisor. Note that this is actually an application of the Identity Property of Multiplication; you are creating an equivalent expression that will have the same answer as the original expression. Special Case: Not enough digits in the dividend to reposition its decimal point (see Model (Move two places also in the dividend. Special Case: The dividend is a whole number (see Model 3 & Model 5 60 Step 4 Bring up the decimal point. Position the decimal point in the quotient directly above the new position of the decimal point in the dividend. To understand why, think about the multiplication of decimal numbers: adding the number of decimal places for both factors gives the number of decimal places in their product. Because division is the inverse operation of multiplication, the number of decimal places in the dividend (including its trailing zeros minus the number of decimal places in the divisor gives the number of decimal places in the resulting quotient

3 Activity 3.4 Dividing Decimal Numbers Steps in the Methodology Example Example 2 Step 5 Divide. Perform the computation and placement of the digits in the quotient as you did with whole numbers. Carry out the division one more place beyond the place value specified for rounding. To do this, add as many trailing zeros in the dividend as are necessary to do the required number of divisions Special Case: Placeholder zeros after decimal point in the quotient (see Model (Note the 2 trailing zeros needed in the dividend to carry the division out 3 places. Step 6 Present the answer. Present your answer by rounding the quotient to the specified place value Answer: Step 7 Validate your answer. Validate: Multiply the non-rounded quotient (from Step 5 by the divisor in its original form. Add the remainder digits of your final subtraction. 3 decimal places 2 decimal places Position the decimal point in your answer. The result must match the original dividend. 5 decimal places = original dividend 6

4 Chapter 3 Decimal Numbers Model Special Case: Not Enough Digits in the Dividend to Reposition its Decimal Point Divide 2.33 by and round the answer to the nearest tenth. Step Set up the division problem Step Step 3 Step 4 One trailing zero is needed to move the decimal point in the dividend If there are not enough digits in the dividend, use as many trailing zeros in the dividend as are necessary to reposition its decimal point. Step To round to tenths, compute the quotient to its hundredths place. Two trailing zeros are needed in the new dividend. Step 6 Round to the tenths place Answer: 64.7 Step 7 Validate: quotient to 2 decimal places (in Step 5 original divisor, 3 decimal places Add the remainder digits of the final subtraction. Properly position the decimal point (5 places. =

5 Activity 3.4 Dividing Decimal Numbers Model 2 Step Special Case: Divisor is a Whole Number Solve and round to the nearest hundredth. Set up the division problem Steps 2, 3 & When the divisor is a whole number, the decimal point is already understood to be after its right-most digit, so it does not move. Therefore, the decimal point in the dividend does not move Step Step Step 7 Validate: Answer: decimal places decimal places decimal places = Model 3 Special Case: Dividend is a Whole Number Divide 4.7 into 62 and round to the nearest hundredth. Step Steps 2, 3 & 4 Step When the dividend is a whole number, place a decimal point to the right of its ones digit and use trailing zero(s to move the decimal point the required number of decimal places. Step Step 7 Validate: Answer: = 62 63

6 Chapter 3 Decimal Numbers Model 4 Special Case: Placeholder Zeros After the Decimal Point in the Quotient Divide: Round to the nearest thousandth. Steps, 2, 3 & Step If the first partial product that the divisor will divide into extends beyond the tenths place of the quotient, you must use zeros as placeholders after the decimal point in the quotient does not divide into 38. Hold the (tenths place with a zero. 52 does not divide into 380. Hold the (hundredths place with a zero. 3 Step Validate: Step Answer: does divide into times = 0.38 Model 5 Step Solve: 8 35 Round to the nearest thousandth Note: Even though 8 < 35, in 8 35, 8 is the dividend and 35 is the divisor. Steps 2, 3, & The decimal point is understood to be to the right of the whole number divisor 35, so it does not move. Therefore, the decimal point in the dividend 8 does not move and remains to the right of Step does not divide into 80. Hold the (tenths place in the quotient with a zero. 35 divides into times. Step Answer: 0.059

7 Activity 3.4 Dividing Decimal Numbers Step 7 Validate: = 8 The technique that follows is a shortcut for dividing decimal numbers by the powers of ten (0, 00, 000, and so on. It is the reverse of the technique used for multiplying decimal numbers by the powers of ten. Technique Move the decimal point to the left as many places as there are zeros in the power of ten. Model Model = = = 48.0 or 48 Move the decimal point one place to the left = 4.80 or =.073 requires a zero placeholder =.0073 requires two zero placeholders =.480 or ,000 = ,000 =.0480 or.048 requires a zero placeholder 65

8 Chapter 3 Decimal Numbers Make Your Own Model Either individually or as a team exercise, create a model demonstrating how to solve the most diffi cult problem you can think of. Answers will vary. Problem: Step Step 5 Step 2 Step 6 Step 3 Step 7 Step 4 66

9 Activity 3.4 Dividing Decimal Numbers. What are three additional situations (other than the context problems for this activity in which you would need to divide decimal numbers? (many possible answers-these are just a few You would use division: if you needed to know how many gallons of gas you bought $23.56/$.399 when fi guring the number of 0.75 pound veal steaks cut from a 2.35 pound rack when fi guring how long it will take you to drive miles at 65.7 miles per hour 2. How do you write a whole number in decimal form? The decimal is implied in all whole numbers and follows the ones digit. 3. What is the shortcut for dividing a number by 0, 00, 000, and so on? The shortcut to dividing by a power of ten is to move the decimal point to the left as many places as there are zeros in the power of ten that you are dividing by. In other words, when dividing by 00, move the decimal point two places to the left. 4. How is dividing decimal numbers different from dividing whole numbers? The difference in the process of dividing decimal numbers and whole numbers is the placement of the decimal point. You also have to determine if the answer is to be rounded at which place value you will need to end the division process. 5. How are the Methodologies for Dividing Decimal Numbers and Dividing Whole Numbers similar? The division of decimals methodology uses the whole number division methodology to perform the actual division. The placement of the decimal is the part that is added to the whole number methodology. Once the decimal placement is fi xed, the division is the same as for whole numbers. 6. How do you determine where the decimal point belongs in the quotient when dividing decimal numbers? The long division methodology is for whole number division; therefore, the divisor must be a whole number. One can move the decimal point in a number by multiplying the number by 0, 00, 000, and so on. Move the decimal point to the right until the divisor is a whole number; move the decimal in the dividend the same number of spaces; that is, multiply the dividend by the same number multiplied by the divisor. If the decimal point is moved the same number of spaces in both the dividend and divisor, then the result has not been changed. The decimal point in the answer is directly above the decimal point in the dividend. (You can do this because of the Identity Property of Multiplication. 7. How do you know when to stop the long division process in a decimal number division problem? The directions will determine when to stop the division process. You need to perform the division to one more place than the directions state. Then you can round to the indicated place value. For example if you are asked to round to hundredths, carry out your division to thousandths place, then round. 67

10 Chapter 3 Decimal Numbers 8. How do you validate division of decimal numbers when you have to round your answer? Rounding the answer requires carrying the division one more place than required. Validate by multiplying the quotient before rounding times the divisor, and then add the remainder. Validate the rounding as usual method or by using the number line and midpoint. 9. What aspect of the model you created is the most difficult to explain to someone else? Explain why. Answers will vary. Fill in the following chart with the correct quotients. Number Perform each division and validate your answers Problem Worked Solution Validation 68

11 Activity 3.4 Dividing Decimal Numbers Problem Worked Solution Validation Round your answer to the nearest hundredth. 4 Divide by 90. Round your answer to the nearest thousandth divides into 526 how many times? Round your answer to the nearest hundredth. 69

12 Chapter 3 Decimal Numbers Problem Worked Solution Validation Round your answer to the nearest thousandth Round your answer to the nearest thousandth. Perform each division, rounding to the decimal place value specified. Validate each answer (nearest tenths (nearest hundredths (nearest hundredths (nearest hundredths (nearest thousandths (nearest thousandths 0.323

13 Activity 3.4 Dividing Decimal Numbers In the second column, identify the error(s you find in each of the following worked solutions. If the answer appears to be correct, validate it in the second column and label it Correct. If the worked solution is incorrect, solve the problem correctly in the third column and validate your answer in the last column. Worked Solution What is Wrong Here? Identify Errors or Validate Correct Process Validation Round the answer to the thousandths place. You have to carry out the division to the tenthousandths place in the quotient to round it to its thousandths place. x (Is the tenthousandths digit 5, <5, or >5? Answer: Round the answer to the hundredths place. 32 does not divide into 5. There needs to be a zero (0 above the 5.. 7

14 Chapter 3 Decimal Numbers Worked Solution What is Wrong Here? Identify Errors or Validate Correct Process Validation Round the answer to the hundredths place. The decimal point in a whole number is to the right of the number Round the answer to the hundredths place. The problem was set up incorrectly. The problem should be