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Written and mental calculations with whole numbers

Module by: Siyavula Uploaders. E-mail the author

MATHEMATICS

Grade 4

WHOLE NUMBERS AND THEIR RELATIONSHIPS

Module 3

WRITTEN AND MENTAL CALCULATIONS WITH WHOLE NUMBERS

Activities:

  • Use a range of techniques to perform written and mental calculations with whole numbers [LO 1.10]
  • Estimate and calculate by selecting and using operations appropriate to solving problems [LO 1.8]
  • Solve problems that involve comparing two or more quantities of the same kind (ratio) [LO 1.7]
  • Use a range of strategies to check solutions and judge their reasonableness [LO 1.11]

FINDING APPROXIMATE ANSWERS AND CHECKING ANSWERS

Now that you have studied “Place Value”, we are going to look at “Rounding Off” numbers so that we can use this to:

  • calculate approximate answers quickly and also
  • check our answers quickly.

1. APPROXIMATING, BY ROUNDING OFF

Consider the following:

1.1 You are riding your bike from your home to the home of a friend who lives 10km away. Your tyre bursts when you have gone 4km. Will you decide to walk home to fix it or go on to your friend?

Yes, you’ll walk home because it’s nearer. 4 is nearer to 0 than to 10

1.2 Now the tyre bursts when you have ridden 6km. Will you decide to walk to your friend’s home or back to your own home?

Yes, you’ll go on to your friend’s home because it is nearer.

6 is nearer to 10 than to 0.

1.3 Now the tyre bursts when you have ridden 5km exactly. Should you decide to walk to your friend’s home or back to your own home?

In Mathematics, always round off upwards if the last digit is 5.

  • Now use the diagrams that we have just seen to help you to complete the table:
Table 1
Number Rounded off to the nearest 10
54  
1 345  
278  
978  
245  
1 133  
684  

1.5 Now we are going to use “rounding off” to calculate, quickly, an approximate answer for the following sums, and then we shall calculate the exact answer, and compare the difference between the two answers. Fill in what is missing in each column:

Table 2
Sum Numbers rounded off to the nearest 10 Approximate answer Exact answer Difference between the 2 answers
24 + 36 20 + 40      
52 + 48 50 + 50      
33 + 52        
79 + 23        
17 + 47        
125 + 46        
411 + 732        

1.6 Look at the sums that you have just completed. In which sums was the approximate answer and the exact answer not very close, and why?

ROUNDING OFF TO THE NEAREST 100:

1.7 Complete the table below. .

Table 3
Number Rounded off to the nearest 100
256  
304  
549  
1 207  
1 399  

ROUNDING OFF TO THE NEAREST 1 000:

1.8 Complete the table below.

Table 4
Number Number rounded off to the nearest 1 000
500  
1 702  
4 089  
723  
1 055  
276  

1.9 Use rounding off to estimate the approximate answer of the following sums. Then calculate the exact answer:

Table 5
SUM with the exact answer. Sum with numbers rounded off to the nearest 10 and the estimated answer:
873 + 46  
   
   
934 － 87  
   
   

2. WORD SUMS

  • Now see how well you can solve word sums without a calculator. Check that your answers are reasonable by rounding off the numbers, but remember that your final answer must be the exact answer. The numbers are not very big and the sums are straightforward, but you will have to read carefully. Write down all you need to write down, and remember to write words with your answer. When you have finished the sums, compare your findings with those of a friend. Enjoy this task.

2.1 In a General Knowledge Competition the Girls’ Team scored 642 points by tea-time. The Boys’ Team scored 493 points. By how many points was the Boys’ Team behind the Girls’ Team?

2.2 By lunch-time the Girls’ Team had 734 points and the Boys’ Team had 655 points.

  1. Was the Boys’ Team catching up?
  2. Why do you say this? Answer carefully.
  3. By how many points was the Boys’ Team behind the Girls’ Team at lunch-time?

2.3 After lunch, the boys made a determined effort. During the afternoon they scored another 619 points. The girls scored 519 points in the afternoon. When all the points were added up, which team eventually won the competition, and by how much?

3. CALCULATOR GAME: two players, one calculator

Figure 1
Figure 1 (Picture 1.png)

  • Continue in this manner. If one of the players makes a mistake, correct it and then the other player gets an extra turn to ask a question. Keep the numbers not more than 4-digit numbers at the most. It’s valuable to become very good at 2-digit numbers first.
  • Complete:

a. 100 – 7 =

b. 1 000 – 7 =

c. 500 – 7 =

d. 500 – 17 =

e. 500 – 27 =

f. 700 – 70 =

g. 1 000 – 70 =

h. 2 100 – 70 =

4. SOME TECHNIQUES to perform written and mental calculations.

  • How can one add 8 + 7 easily?

Figure 2
Figure 2 (Picture 2.png)

Figure 3
Figure 3 (Picture 3.png)

4.2 Discuss: which learner was right? What method must you use?

  • You must use the method that you understand best, the one that you feel comfortable with, and you must also try to listen to others when they explain their methods. But be sure to use the method that you really understand well enough to explain to others what you did.4.3 Try to see a link between these sums as you write down the answers

a. 8 + 7 =

b. 18 + 7 =

c. 8 + 17 =

d. 18 + 17 =

e. 8 – 7 =

f. 18 – 7 =

g. 28 – 7 =

h. 28 – 17 =

5. Now use your method and try some written sums. Write down all the steps you needed to reach the answer. You may not use a calculator.

  • 87 － 54
  • 84 － 57
  • Now discuss these two sums and their answers with a friend.
  • Explain what you noticed.

6. Now calculate, without a calculator and using the method that you feel you understand most. Write down all the steps of your calculation:

  • 1 345 + 278
  • 978 – 245
  • 1 278 + 1 133
  • 845 – 672
  • 684 – 659
  • 4 092 + 3 214
  • Check the last sum by rounding off the numbers to the nearest 10 or nearest 100 and then calculating an approximate answer. Then discuss how you reached your answers with a friend. If necessary, check your answers on a calculator.

7. MORE WORD SUMS

Sales at a Craft Market for the first 5 months of the year:

Table 6
Months Cooldrinks Hot dogs Ice-creams Mugs of Soup
January 3 064 1 754 2 356 225
February 3 215 1 036 2978 54
March 1 964 2 375 2 035 987
April 874 3 752 1 096 1 952
May 756 3 904 788 2 659

7.1 How many cooldrinks were sold altogether during the five months?

7.2 Were cooldrinks or ice-creams more popular during the five months? Explain why you give this answer.

7.3 Cooldrinks cost R5,00 each. How much money was collected for cooldrinks in May? Try to find an easy way of calculating this and write it down.

7.4 At the beginning of January the ice-cream stall holder buys 24 boxes of ice-creams. Each box contains 100 ice-creams. How many ice-creams are over at the end of the January Market?

7.5 Which month was the coldest? Why do you say so? (Look back at the table showing the sales.)

7.6 Round off the numbers of cups of soup to the nearest 100 and say approximately how many cups of soup were sold altogether.

Assessment

Table 7
Learning outcomes(LOs)
 
LO 1
Numbers, Operations and RelationshipsThe learner will be able to recognise, describe and represent numbers and their relationships, and to count, estimate, calculate and check with competence and confidence in solving problems.
Assessment standards(ASs)
 
We know this when the learner:
1.7 solves problems that involve:
  • comparing two or more quantities of the same kind (ratio).
1.8 estimates and calculates by selecting and using operations appropriate to solving problems that involve:
  • rounding off to the nearest 10; 100 or 1 000;
1.9 performs mental calculations involving: addition and subtraction:
  • adding and subtraction;
1.10 uses a range of techniques to perform written and mental calculations with whole numbers including:
  • building up and breaking down numbers;
  • rounding off and compensating;
  • doubling and halving;
  • using a number-line;
  • using a calculator.
1.11 uses a range of strategies to check solutions and judge the reasonableness of solutions.

Memorandum

ACTIVITY

1.1 home

1.2 friend’s home

1.3 either / friend’s home

1.4 50; 1 350; 280; 980; 250; 1 130; 680

1.5

Table 8
Sum Numbers rounded off to nearest 10 Approximate answer Exact answer Difference
24 + 36 20 + 40 60 60 0
52 + 48 50 + 50 100 100 0
33 + 52 30 + 50 80 85 5
79+ 23 80 + 20 100 102 2
17 + 47 20 + 50 70 64 6
125 + 46 130 + 50 180 171 9
411 + 732 410 + 730 1 140 1 143 3

1.6 They are close when one number is rounded off upwards and the other, downwards. When both numbers are rounded off upwards, or both are rounded off downwards the totals are not very close, e.g. the third sum; the fifth sum and the last two sums. The rounding off increases the gap.

1.7 300; 300; 500; 1 200; 1 400

1.8 1 000; 2 000; 4 000; 1 000; 1 000; 0

  • 873 + 46 = 919;

870 + 50 = 920

934 – 87 = 847;

930 – 90 = 840

2 WORD SUMS

2.1 642 – 493 =149; 640 – 490 = 150

The boys were behind by 149 points.

2.2 (a) Yes

(b) 734 – 655 =79; 730 – 660 = 70

At tea-time the difference between the Girls’ points and the boys’ was 149 points; at lunch-time the difference was only 79 points, so the boys were catching up.

(c) 79 points, see above

(d) Girls

734 + 519 = 1 253

Boys

655 + 619 = 1 274

The boys won by 21 points.

3.1 Calculator Game

3.2 (a) 93

(b) 993

(c) 493

(d) 483

(e) 473

(f) 630

(g) 930

(h) 2 030

4.1 and 4.2 Discussion: Techniques

4.3 (a) 15

(b) 25

(c) 25

(d) 35

(e) 1

(f) 11

(g) 21

(h) 11

5. WRITTEN SUMS

5.1 33

5.2 27

5.3 and 5.4 discussion and explanation

6.1 1623

6.2 733

6.3 2 411

6.4 173

6.5 25

6.6 7 306

6.7 Checking by rounding off and discussion

  • 9 873 cool drinks
  • Cool drinks; 620 more cool drinks were sold than ice-creams
  • 44 ice-creams were over
  • Hot dogs
  • May; many hot-dogs and mugs of soup were sold; few cool drinks and ice creams were sold.
  • 6 000 cups of soup

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