Eg Adding and Subtracting Strategies

Using approximate numbers EA

Formal column rounding AA We are learning to round numbers to a specified column value. AM

Notes AP 1) In mathematics, there is sometimes a need to round numbers or answers, because of this, formal methods of rounding numbers have been created

2) The simplest method of rounding is called column rounding, as you round numbers by looking at the columns you want to keep

3) Formal rounding methods work in the same way as the rounding methods we have used before, we just explain them differently

for example To round a number to the nearest ten,

Look at the number in the next column along from the tens column (the ones column)

If the number is a 0, 1, 2, 3, or 4 If the number is a 5, 6, 7, 8, or 9 Put a zero in the ones column Add one to the tens column then put a zero in the ones column

examples (1) round 374 to the nearest ten

374  370  374 rounds to 370

(2) round 1035 to the nearest ten

1035  1040  1035 rounds to 1040 Exercise 1 Round these numbers to the nearest ten 1) 17 (2) 306 (3) 182 (4) 4155 (5) 164

6) 2091 7) 13427 (8) 304,296 (9) 359,214 (10) 966,355 Notes The formal methods of rounding work whatever column you decide to round to

for example (1) Round 12345 to the nearest hundred

Find the number in the hundreds column (the 3), and decide whether the number rounds to 12300 or 12400

12345  12300  as there is a 4 in the tens column we round down

(2) Round 201,936 to the nearest thousand

Find the number in the thousands column (the 1) and decide which thousand the number rounds to

201,936  202,000  201,936 rounds to 202,000 (as there is a 9 in the next column, we round up) Exercise 2 Round these numbers to the nearest hundred 1) 352 (2) 179 (3) 2134 (4) 2037 (5) 1599

6) 7135 (7) 8061 (8) 5938 (9) 7717 (10) 10263 Exercise 3 Round these numbers to the column asked for 1) Round 74 to the nearest ten (2) Round 134 to the nearest ten

3) Round 392 to the nearest hundred (4) Round 476 to the nearest ten

5) Round 1091 to the nearest thousand (6) Round 2399 to the nearest hundred

7) Round 7346 to the nearest ten (8) Round 9147 to the nearest thousand

9) Round 2155 to the nearest ten (10) Round 4652 to the nearest hundred

11) Round 777 to the nearest hundred (12) Round 469 to the nearest ten

13) Round 13,436 to the nearest ten thousand (14) Round 39,129 to the nearest thousand

15) Round 299 to the nearest ten (16) Round 3861 to the nearest hundred

17) Round 804 to the nearest thousand (18) Round 83 to the nearest hundred One-figure checks We are learning to use approximate numbers to roughly work out the size of the answer for multiplication problems.

Calculators DON’T always get the right answer  you may push the wrong button  you may have dropped it and damaged it  the batteries may be too weak for it to work properly So when you use a calculator you should always check the answer in your head to see that it is about right

for example 79  31 = 217 (according to a calculator)

To check this, work out a similar problem that you can do easily in your head 79 is close to 80 (which is written as 79  80) 31 is close to 30 (31  30)

so 79  31  80  30  2400 217 is not near 2400 so the answer was wrong

2) Doing a problem like this is called doing a mental check  if you can’t do it in your head it is too hard - try easier numbers  in it takes longer than, say, 10 seconds it is also too hard A mental check is supposed to be an easy, painless way of making sure that your answers make sense or are about right when you are doing problems - especially in a test

3) In this problem we have rounded the numbers so there was only one figure other than zeros. When you do this so you can work out a rough answer in your head it is called doing a one-figure check. A question to discuss How does the answer to a one-figure check tell us whether or not the answer to a problem ‘is about right’?

More examples of one figure checks 1) do a one-figure check for 86  357 86  90 357  400 so 86  357  90  400  36000

2) do a one-figure check for 8  49 8 is already one figure - so don’t change it 49  50 so 8  49  8  50  400 Exercise 4 Do one-figure checks for these problems. Initially do them in your head, then write down your checks on paper. 1) 36  57 (2) 55  82 (3) 56  34 (4) 707  43

5) 92  87 (6) 291  477 (7) 8062  154 (8) 146  361

9) 9  57 (10) 6  42 (11) 8  29 (12) 3  426

13) 352  88 (14) 261  305 (15) 1407  3 (16) 1213  8

17) 482  235 (18) 1229  403 (19) 511  811 (20) 337  523

Using known fact checks We are learning to use approximate numbers to roughly work out the size of the answer for multiplication problems.

Sometimes when working out an approximate answer to a problem, it is better not to round all the numbers to one figure. For example, when finding an approximate answer to 108  423, there are a number of sensible ways to approximate the answer

Method 1 Method 2 Method 3 (one figure check) (Known facts) (Known facts) 108  100 108  110 108  100 423  400 so 423  400 so 423 = 423 so 108  423  100  400 108  423  110  400 108  423  100  423  40000  44000  42300 Some things to discuss  Which method gives the best approximation of the answer?  How would you know beforehand which method would give the best approximation? Remember that the purpose of finding an approximate answer is to confirm mentally that the real answer ‘seems right’. Exercise 5 Work out approximate answers to these problems using both a one-figure check and a known fact check. Also do the problems on a calculator and say which method gave you the closest estimate of the answer 1) 12  135 (2) 15  31 (3) 106  83 (4) 47  153

5) 259  19 (6) 196  182 (7) 305  1453 (8) 134  29

9) 89  108 (10) 205  111 (11) 307  321 (12) 809  125

13) 1350  125 (14) 1271  33 (15) 1492  321 (16) 809  125

17) Be prepared to discuss what you have done with each question, and what you think is the best way to find an approximate answer. This will involve you going back to questions 1 to 16 and making a decision about which is the more accurate of the two methods in each case, and why it is better in this case. Overall, try to make up some guidelines on when each method is useful.

Up and down checks We are learning to use approximate numbers to roughly work out the size of the answer for multiplication problems.

Sometimes when working out an approximate answer to a problem, it is better not to round all the numbers to the closest tidy number

For example You want to work out an approximate answer to 29  26

Try the following approximations 30  30 30  26 30  25 Some things to discuss  How do you think the third strategy works?  Which approximation of the answer was ‘the best’?  How can you work out when an ‘up and down’ strategy would be worthwhile using? Exercise 6 Work out approximate answers to these problems using the ‘up and down’ method. Try using one other strategy for each question to help you get a ‘feel’ for when this strategy is useful. Use a calculator to work out the exact answers 1) 16  47 (2) 87  31 (3) 106  83 (4) 44  126

5) 155  19 (6) 196  185 (7) 216  35 (8) 389  211

9) 89  106 (10) 305  111 (11) 306  326 (12) 809  125

13) 126  38 (14) 26  58 (15) 254  19 (16) 324  223

17) Be prepared to discuss what you have done with each question, and what you think is the best way to find an approximate answer to each question. As in exercise 5, this will involve you going back to questions 1 to 16 and making a decision about what is the more accurate of the two methods in each case, and continuing to develop your ideas about when to use particular rounding strategies. Exercise 7 Work out approximate answers to each of these questions. Use the most appropriate method for each. For each question explain why you think it is the best method 1) 23  67 (2) 45  91 (3) 123  41 (4) 27  52

5) 142  64 (6) 362  185 (7) 441  266 (8) 739  203

9) 37  111 (10) 283  215 (11) 458  116 (12) 234  234

13) 327  381 (14) 39  77 (15) 899  219 (16) 546 388

Exercise 8 You can find approximate answers for problems other than multiplications. Use one of the methods you have learned to estimate these answers in your head. Write down what you did for each so you can discuss your strategies with others later. 1) 1154 + 129 (2) 2198 - 855 (3) 479  49 (4) 1038 + 924

5) 1359 - 138 (6) 407  18 (7) 1437  714 (8) 1536 - 395

9) 1482 + 3075 (10) 891  34 (11) 2627 - 374 (12) 26109 + 5174 Using approximate numbers Answers Exercise 1 1) 17  20 (2) 306  310 (3) 182  180 4) 4155  4160 (5) 164  160 (6) 2091  2090 7) 13,427  13,430 (8) 304,296  304,300 (9) 359,214  359,210 10) 966,355  966,360 Exercise 2 1) 352  400 (2) 179  200 (3) 2134  2100 (4) 2037  2000 5) 1599  1600 (6) 7135  7100 (7) 8061  8100 (8) 5938  5900 9) 7717  7700 (10) 10,263  10,300 Exercise 3 1) 74 to the nearest ten is 70 2) 134 to the nearest ten is 130 3) 392 to the nearest hundred is 400 4) 476 to the nearest ten is 480 5) 1091 to the nearest thousand is 1000 6) 2399 to the nearest hundred is 2400 7) 7346 to the nearest ten is 7350 8) 9147 to the nearest thousand is 9000 9) 2155 to the nearest ten 2160 10) 4652 to the nearest hundred is 4700 11) 777 to the nearest hundred is 800 12) 469 to the nearest ten is 470 13) 13,436 to the nearest ten thousand is 10,000 14) 39,129 to the nearest thousand is 39,000 15) 299 to the nearest ten is 300 16) 3861 to the nearest hundred is 3900 17) 804 to the nearest thousand is 1000 18) 83 to the nearest hundred is 100

A question to discuss By leaving the biggest figure in each number, a one-figure check lets us create an approximate answer ‘of about the right size’. We talk of this formally as being ‘of the correct magnitude’) Exercise 4 1) 36  57  40  60 (2) 55  82  60  80  2400  4800 3) 56  34  60  30 4) 707  43  700  40  1800  2800 5) 92  87  90  90 (6) 291  477  300  500  8100  150,000 7) 8062  154  8000  200 (8) 146  361  100  400  1,600,000  40,000 9) 9  57  9  60 (10) 6  42  6  40  540  240 11) 8  29  8  30 (12) 3  426  3  400  240  1200 13) 352  88  400  90 (14) 261  305  300  300  36,000  90,000 15) 1407  3  1000  3 (16) 1213  8  1000  8  3000  8000 17) 482  235  500  200 (18) 1229  403  1000  400  100,000  400,000 19) 511  811  500  800 (20) 337  523  300  500  400,000  150,000 Some things to discuss  Which method gives the best approximation of the answer? The method that gives the better approximation of the answer often depends on the numbers in the problem. Sometimes a one-figure check is a really good method, at others, a basic fact check is best. You heave to develop ‘approximation sense’, that is get a feel for what will be best in which case.  How would you know beforehand which method would give the best approximation? By looking at the numbers, and checking which method would keep the numbers as close as possible to the original ones. Exercises 5 to 8 Student responses to these exercises will vary, and need to be discussed with others to help develop ‘approximation sense’. Some things to discuss  How do you think the third strategy works? By rounding on number up, and the other down, we are creating a form of balance, where what happens to the numbers overall is not changed too much. This allows for a more accurate approximation  Which approximation of the answer was ‘the best’? Again, this depends on the numbers in the question. You may also have noticed that sometimes a one-figure check also works like an up and down check, and at other times the other methods seem to blend to give you a feel for what will give ‘the best’ approximation  How can you work out when an ‘up and down’ strategy would be worthwhile using? This has been answered in the answer to the first question