Approximations - Rounding Mathematics Approximations - Rounding

The aim of this powerpoint is to teach you techniques for approximating numbers. EITHER Take notes as you go along, include some examples and write down any questions and your answers (which you can mark as you go along) OR At the end of the powerpoint, printout the notes called Calc2a

Approximations To approximate means to give a rough estimate, not an exact value. The symbol ‘≈’ means approximately equal to. It is useful to be able to approximate (or ‘round’ values up or down) as one method for checking that answers to calculations are sensible as well as make values easier to work with.

Rounding – Method 1 Look at the degree of rounding required. Imagine the two rounded values that lie either side of the actual value quoted on a number line. Which of the two values is the actual value closest to? If the actual value is exactly half-way between them, then round UP.

Example 1 Round 76 to the nearest ten. The two ‘tens’ values either side of 76 are 70 and 80. Which of the two values is the actual value closest to? 75 70 76 80 ANS: 76 is closest to 80.

Example 2 Round 1949 to the nearest hundred. The two ‘hundred’ values either side are 1900 and 2000. Which of the two values is the actual value closest to? 1950 1900 1949 2000 ANS: 1949 is closest to 1900.

Example 3 Round 21500 to the nearest thousand. The two ‘thousand’ values either side are 21000 and 22000. Which of the two values is the actual value closest to? 21500 21000 21500 22000 ANS: 21500 is exactly halfway so round UP to 22000.

Rounding – Method 2 Find the digits up to and including the ‘rounding’ column. If the number on the right of these digits is 5 or more, add 1 on to these digits. If the number on the right of these digits is 4 or less, leave the digits as they are. All the other digits on the right of these now become zeros (or ignored if they are zeros at the end after the decimal point).

Example 1 Round 120.63 to the nearest whole number. (i.e. unit) The digits up to and including the units column are 120. The number on the right is 6 (which is 5 or more) so add 1 on to the ‘120’ to get ‘121’ The digits on the right now become zeros (and can be ignored as they are after the decimal point). ANS: 120.63 rounds to 121 (to the nearest whole number)

Example 2 Round 1749.7 to the nearest hundred. The digits up to and including the hundreds column are 17. The number on the right is 4 (which is LESS than 5) so 17 stays as it is… The digits on the right now become zeros (and any zeros after the decimal point can be ignored). ANS: 1749.7 rounds to 1700 (to the nearest hundred)

Example 3 Round 5.9718 to the nearest tenth The digits up to and including the tenths column are 5(.)9. The number on the right is 7 (which is 5 or more) so add 1 on to the ‘5(.)9’ to get ‘6(.)0’ The digits on the right now become zeros. ANS: 5.9718 rounds to 6.0 (to the nearest tenth) You could quote the answer as 6, but it is good practice to include the tenths value (even though it is 0) as you were asked to round the value to the nearest tenth.

Practice Questions Work out the answers to each of these questions before moving on to the next slides to check them. Q1. Round each of these values to the nearest 10. a) 753 ≈ b) 1378 ≈ c) 5702 ≈ Q2. Round each of these values to the nearest 100. a) 1838 ≈ b) 23750 ≈ c) 12792 ≈ Q3. Round each of these values to the nearest 1000. a) 48753 ≈ b) 32178 ≈ c) 129609 ≈

Practice Question 1 Round each of these values to the nearest 10. 753 ≈ 1378 ≈ 5702 ≈ Between 750 – 760 753 is closest to 750 Is it 1370 or 1380? 1378 is closest to 1380 ‘570’ are the digits up to and including the tens 2 is on the right but is less then 5 570 stays as it is but the 2 becomes a zero 5702 is closest to 5700 NB. Rounding to 10 means the answer must end in ‘0’

Practice Question 2 Round each of these values to the nearest 100. 1838 ≈ 23750 ≈ 12792 ≈ Between 1800–1900 1838 is closest to 1800 Is it 23700 or 23800? 23750 rounds up to 23800 ‘127’ are the digits up to and including the hundreds 9 on the right is more than 5 so… …add 1 on to 127 …then the ’92’ become zeros 12792 is closest to 12800 NB. Rounding to 100 means the answer must end in ‘00’

Practice Question 3 Round each of these values to the nearest 1000. 48753 ≈ 32178 ≈ 129609 ≈ Between 48000–49000 48753 is closest to 49000 Is it 32000 or 33000? 32178 rounds to 32000 ‘129’ are the digits up to and including the thousands 6 on the right is more than 5 so… …add 1 on to 129 to get 130 …then the ’609’ become zeros 129609 is closest to 130000 NB. Rounding to 1000 means the answer must end in ‘000’

More Practice Questions Work out the answers to each of these questions before moving on to the next slides to check them. Q4. Round each of these values to the nearest unit. a) 8.63 ≈ b) 12.15 ≈ c) 0.499 ≈ Q5. Round each of these values to the nearest tenth. a) 0.8501 ≈ b) 10.449 ≈ c) 3.083 ≈

Practice Question 4 Round each of these values to the nearest unit. 8.63 ≈ 12.15 ≈ 0.499 ≈ Between 8 – 9 8.63 is closest to 9 Is it 12 or 13? 12.15 is closest to 12 ‘0’ is the digit up to and including the units 4 is on the right but is less then 5 0 stays as it is but the ‘499’ become zeros (and as they all come after the decimal point they can be ignored) 0.499 is closest to 0.000  0

Practice Question 5 Round each of these values to the nearest tenth. 0.8501 ≈ 10.449 ≈ 3.083 ≈ Between 0.8 – 0.9 0.8501 is closest to 0.9 Is it 10.4 or 10.5? 10.449 is closest to 10.4 ‘3.0’ are the digits up to and including the tenths 8 on the right is more then 5 so… …add 1 on to the (3.)0 to get (3.)1 …then the ‘83’ become zeros and can be ignored 3.083 is closest to 3.1

What next? (Page 1) If you haven’t made any notes or copied any examples, questions and answers out during this presentation, print out the notes called Calc2a. Read through them and make sure you answer any questions. Work through pages 5 to 9 of the MyMaths lesson called Estimating Amounts found at: http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=estimating/estimatingAmounts&taskID=1219 Work through the MyMaths lesson (and then the online homework) called Rounding to 10, 100 found at: http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=accuracy/rounding10&taskID=1003 http://www.mymaths.co.uk/tasks/library/loadTask.asp?title=accuracy/rounding10OH&taskID=1003 Continued on the next page…

What next? (Page 2) Please work through pages 1 to 4 (making notes and copying examples as well as questions and answers) of the MyMaths lesson (and then the online homework) called Estimating Amounts at: http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=estimating/estimatingAmounts&taskID=1219 http://www.mymaths.co.uk/tasks/library/loadTask.asp?title=estimating/estimatingAmountsOH&taskID=1219 To recognise when you might need to round an answer up or down according to the situation, please work through (making notes and copying examples as well as questions and answers) the MyMaths lesson (and then the online homework) called Solving Problems by Rounding found at: http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=estimating/estimatingGold&taskID=1373 http://www.mymaths.co.uk/tasks/library/loadTask.asp?title=estimating/estimatingGoldOH&taskID=1373 Save and complete the worksheets called Round10-100.xlsx and Round-S1.xlsx Now move on to the Calc2b-dp-sf powerpoint