complete the arithmetic test as quickly and accurately as possible. You have only three minutes!
a) 17 + 9 = _________________
b) 57 + 9 = _________________
c) 45 – 11 = _________________
d) 65 - 21 = _________________
e) Double 47: _________________
f) 17 + 19 + 13 = _________________
g) 23 + 19 + 7 = _________________
h) 210 + 120 + 21 = _________________
i) Double 184: _________________
j) 10 000 – 16 = _________________
k) Halve 64: _________________
l) Halve 96: _________________
m) 6 × 8 = _________________
n) 54 ÷ 9 = _________________
o) One quarter of 96 = _________________
p) 12 × 8 = _________________
q) _________________ ÷ 12 = 11
r) 63 ÷ _________________ = 9
s) (15 ÷ 15) × 15 = _________________
t) (56 + 63) + 44 = 56 + (_________________ + 63)
Table 1.14.
HOW DID YOU FARE?
I had all right!
MARK THE RELEVANT BOX: I had more than half right!
I did not do well enough.
5. Now let us add even greater numbers. See if you can complete the following neatly and
correctly. Start at 33 000 and add 10 000 at a time. Build yourself a "footpath" by colouring every block with the correct answer green.
Table 1.15.
33 000 40 000 58 000 85 000 93 000
102 000 110 000 133 000
38 000 43 000 60 000 83 000 96 000
103 000 111 000 123 000
41 000 53 000 62 000 73 000 99 000
107 000 113 000 121 000
32 000 55 000 63 000 71 000 100 000 109 000 116 000 119 000
DO YOU STILL REMEMBER?
If we want to programme the calculator to check the previous sum, we key in 33 000 + 10 000 =
= =.
Use your calculator to check whether your “footpath” has been calculated correctly.
Assessment
Learning Outcome 1: The 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 Standard 1.1: We know this when the learner counts forwards and backwards in
whole number intervals and fractions;
Assessment Standard 1.8: We know this when the learner estimates and calculates by selecting and using operations appropriate to solving problems;
Assessment Standard 1.9: We know this when the learner We know this when the learner
performs mental calculations.
1.14. To use a series of techniques to do calculations*
1.14. To use a series of techniques to do calculations*
MATHEMATICS
Number Concept, Addition and Subtraction
Addition
EDUCATOR SECTION
Memorandum
1. 262; 364; 466; 568; 670; 772; 874; 976; 1 078
2.
2.1 2 589; 2 592; 2 596; 2 598; 2 601; 2 604; 2 607
2.2 7 939; 8 039; 8 139; 8 239; 8 339; 8 439; 8 539
2.3 18 213; 28 213; 38 213; 48 213’ 58 213; 68 213; 78 213; 88 213
Leaner Section
Content
Activity: To use a series of techniques to do calculations [LO 1.10]
1. The calculator is a wonderful aid for checking our calculations. If you know how to use the
"constant function" of your calculator, an activity like the previous and following ones is child's play! See if you can complete the following arrow diagram with the aid of your calculator.
What are you going to key in? ______________________________________
Figure 1.35.
2. Use the constant function of your calculator again and write down the numbers that appear on the screen when you key in the following:
2.1 2 586 + 3 = = = 2 607
__________________________________________________________________________________________________________________________________________
2.2 7 839 + 100 = = = 8 539
__________________________________________________________________________________________________________________________________________
2.3 8 213 + 10 000 = = = 88 213
__________________________________________________________________________________________________________________________________________
You know by now that different peoples used different numerals to write down numbers. Let us
take an in-depth look at the Romans' way of writing:
DO YOU STILL REMEMBER?
In the previous module we saw that the Roman numerals differed from ours.
Table 1.16.
Our numbers:
1 2 3
4 5 6 7
8
9 10
Roman numerals: i ii iii iv v vi vii viii ix x
Assessment
Learning Outcome 1: The 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 Standard 1.10: We know this when the learner uses a range of techniques to perform written and mental calculations with whole numbers.
1.15. To describe and illustrate different cultures' ways
of writing*
MATHEMATICS
Number Concept, Addition and Subtraction
Addition
EDUCATOR SECTION
Memorandum
1.1 xi
1.2 xv
1.3 watches
2.
2.1 XLII
2.2 XCIX
2.3 MMDL
3.
3.1 608
3.2 65
3.3 3 257
4. Will differ from year to year
Brain Teaser
1. Move “minus” to just before “x” so that sum reads 1 x 1 = 1
2. Take one away from = and add to – so that sum reads 1 = 111 – 11
Leaner Section
Content
Activity: To describe and illustrate different cultures' ways of writing [LO 1.2]
Can you deduce from this how the Romans would have written 11?
_____________________________________________________________________
1.2 How would they have written 15?
_____________________________________________________________________
1.3 Where are these Roman numerals still used nowadays?
_____________________________________________________________________
Look carefully at the following:
Table 1.17.
We:
10
20
30
40
50
60
70
80
90
100
The Romans: X
XX XXX XL L
LX LXX LXXX XC C
We:
100 200 300
400 500 600 700
800
900 1 000
The Romans: C
CC CCC CD D
DC DCC DCCC CM M
2. Use the information given above and calculate the following.
Write your answers in Roman numerals:
40 + 2 ________________________________________________________
2.2 90 + 9 ________________________________________________________
2.3 2 000 + 500 + 50 _______________________________________________
3. You have just practised writing like the Romans. Below are more examples of how they would
write certain numbers. Can you write them in ordinary (our) numbers?
3.1 DCVIII ________________________________________________________
3.2 LXV ________________________________________________________
3.3 MMMCCLVII __________________________________________________
4. Write our present date in Roman numerals.
BRAIN TEASER
The matches below have been set out in Roman numerals, but the answers are incorrect. Can you
“fix” each sum, so that the answer is correct, by moving only ONE match?
Figure 1.36.
TIME FOR SELF-ASSESSMENT
Evaluate yourself on a scale of 1 - 4 by just drawing a circle around the figure that is true about yourself.
Table 1.18.
Not at
Some-
Most of the
I can
Always
all
times
time
Use the constant function of my pocket calculator for
1
2
3
4
repeated addition
Write down Roman numerals for our numbers
1
2
3
4
Add and give the answer in Roman numerals
1
2
3
4
Write Roman numerals in ordinary numbers
1
2
3
4
However, the Roman counting system gets very complicated. The decimal system (OR METRIC
SYSTEM) that we use is more practical. Let us consider the different ways in which we can add.
DO YOU REMEMBER?
The answer of an addition sum is called the SUM OF THE TWO NUMBERS.
The INVERSE (reverse) calculation of addition is SUBTRACTION.
DID YOU KNOW?
The order in which numbers are added makes no difference to the answer.
Table 1.19.
For example: 5 + 4 = 4 + 5
9 = 9
Table 1.20.
As well as: (16 + 14) + 12 = 16 + (14 + 12)
30 + 12 = 16 + 26
42 = 42
Assessment
Learning Outcome 1: The 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 Standard 1.2: We know this when the learner describes and illustrates various ways of writing numbers in different cultures (including local) throughout history.
1.16. To recognise, describe and use laws*
MATHEMATICS
Number Concept, Addition and Subtraction
Addition
EDUCATOR SECTION
Memorandum
1.
1.1 9 + (8 + 7) = 24
1.2 121 + 32 = 153 / 140 + 13 = 153
1.3 338 + 100 = 438 / 396 + 42 = 438
Leaner Section
Content
Activity: To recognise, describe and use laws [LO 1.12]
1. Just as every country has certain laws to ensure good governance, there are "laws" or
"characteristics" in Mathematics that can help you calculate answers more easily. There are, for example, the commutative and associative characteristics of addition. Complete the following and see how they work!
1.1 (9 + 8) + 7 = 9 + _______________ = _______________
1.2 121 + (19 + 13) = 121 + _______________ = _______________
(121 + 19) + 13 = _______________ + 13 = _______________
1.3 338 + (58 + 42) = 338 + _______________ = _______________
(338 + 58) + 42 = _______________ + 42 = _______________
1.4 Discuss with your classmates: Which grouping of addition numbers was the easiest? Why?
DID YOU KNOW
When we want to add numbers, we can calculate the answer quickly by ROUNDING OFF the
numbers!
Assessment
Learning Outcome 1: The 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 Standard 1.12: We know this when the learner recognises, describes and uses.
1.12.3 the commutative, associative and distributive properties with whole numbers (the
expectation is that learners should be able to use the properties, and not necessarily know the names.
1.17. To estimate and calculate by selecting and using
suitable operations*
MATHEMATICS
Number Concept, Addition and Subtraction
Addition
EDUCATOR SECTION
Memorandum
1. Look at the unit’s digit.
If it is 5 or more, the tens digit becomes one more.
If it is 4 or less, the tens digit remains the same.
2. Look at the tens digit.
If it is 5 or more, the hundreds digit becomes one more.
If it is 4 or less, the hundreds digit remains unchanged.
3. Look at the hundreds digit.
If it is 5 or more, the thousands digit becomes one more.
If it is 4 or less, the thousands digit remains the same.
2.
2.1 20 + 10 + 20 = 50
2.2 30 + 10 + 50 = 90
2.3 10 + 20 + 20 = 50
2.4 100 + 200 + 300 = 600
2.5 200 + 100 + 100 = 400
3. “Estimated answer” and “Difference” will be different as a result of the method
applied. Pocket calculator gives correct answer.
4.
a) 279
b) 1 660
c) 6 581
d) 2 843
5. Answers will vary – many combinations possible
6. 35 + 15; 47 + 3; 23 + 27
50 + 150; 25 + 175; 10 + 190; 180 + 20; 170 + 30
250 + 750; 400 + 600; 900 + 100; 200 + 800; 990 + 10
6. 250 + 750; 125 + 825; 950 + 50; 450 + 550; 225 + 775; 415 + 585
7. 5 000 + 400 + 80 + + 9 + 300 + 60 + 2 + 1 000 + 500 + 70
= (5 000 + 1 000) + (400 + 300 + 500) + (80 + 60 + 70 +) + (9 + 2)
= 6 000 + 1 200 + 210 + 11
= 7 421
Leaner Section
Content
Activity: To estimate and calculate by selecting and using suitable operations
[LO 1.8]
Now practise your skill at estimating with confidence the answers by rounding off numbers.
1. It is important that we consider once again what rounding off is, and how it is done. Explain to a friend how we round off to the nearest 10, 100 and 1 000.
2. Now work in groups of three and see who is able to calculate the answers without a pocket
calculator by rounding off to the closest 10 or 100. (You may not write anything down!)
2.1 16 + 11 + 19
2.2 34 + 9 + 52
2.3 2 + 15 + 23
2.4 141 + 208 + 339
2.5 182 + 149 + 106
3. Look carefully at the following example:
32 + 87 + 96 + 42 becomes
30 + 90 + 100 + 40 (rounded off to the closest 10)
Therefore the answer is approximately 260.
When we want to add larger numbers, we can round off to the closest 100:
138 + 759 + 304 becomes
100 + 800 + 300
Therefore the answer will be roughly 1 200.
DID YOU KNOW?
We can also round numbers off to a lower or upper limit!
e.g. 62 + 75 + 94 + 45
Lower limit: 60 + 70 + 90 + 40 = 260
Upper limit: 70 + 80 + 100 + 50 = 300
The correct answer will therefore be between
260 and 300.
4. Now complete the table by rounding off the numbers according to the method that you prefer.
Estimate the answer first. Then calculate the correct answer with a pocket calculator. Finally write down the difference between the two answers.
Table 1.21.
Calculation
Estimated answer Calculator Difference
E.g. 308 + 475
300 + 480 = 780 783
3
a)
67 + 98 + 114
b)
896 + 723 + 41
c)
1 892 + 4 689
d)
2 458 + 294 + 91
TIME FOR SELF-ASSESSMENT!
Colour the correct box:
Table 1.22.
I am able to add by rounding off to the closest 10, 100 or 1
** have not mastered
mastered
000
yet
** Ask your educator to give you some more examples that you can do at home!!
5. There are also other techniques or computations (other than rounding off) that you can use to calculate answers quickly. Look carefully at the examples and try to add more number
combinations.
Figure 1.37.
6. Look at the numbers in the diagram below. By finding as many as possible addition strings that give a sum of 1 000, you can calculate the sum of all the numbers in a jiffy! How many do you
see? Colour them in the same colour.
Table 1.23.
250
585
125
950
750 825 50
550 775 415
450
225
316
775
415
DO YOU STILL REMEMBER?
We can use grouping to add up. Look carefully at the example!
= 370 + 245 + 684
= 300 + 70 + 200 + 40 + 5 + 600 + 80 + 4
= (300 + 200 + 600) + (70 + 40 + 80) + (5 + 4)
= 1 100 + 190 + 9
= 1 299
7. Use this method to calculate:
5 489 + 362 + 1 570
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
Assessment
Learning Outcome 1: The 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 Standard 1.8: We know this when the learner estimates and calculates by selecting and using operations appropriate to solving problems.
1.18. To solve problems in context*
MATHEMATICS
Number Concept, Addition and Subtraction
Addition
EDUCATOR SECTION
Memorandum
Leaner Section
Content
Activity: To solve problems in context [LO 1.6]
To use a series of techniques to do calculations [LO 1.10]
To use strategies to check solutions [LO 1.11]
1. We have now investigated a few techniques to calculate answers. We encounter problems like
those below every day. If we are unable to solve them, we will not be able to do calculations with much confidence! Form groups of three and try to solve the following problems. Your educator
will supply the paper you need.
1.1 Dimitri and his friends decided to go to the “Bites-and-Bits” restaurant.
The menu is as follows:
Sandwiches:
Cheese R9,99
Ham and cheese R12,99
Egg R11,99
Tomato, ham and cheese R15,99
Chicken R15,99
Beverages:
Coke, Fanta, Sprite, Cream Soda R4,95
Milkshake: Strawberry, Banana, Chocolate R8,95
Dimitri orders a ham and cheese sandwich and a strawberry milkshake.
Vusi orders a chicken sandwich and a Coke.
Nancy orders an egg sandwich and a chocolate milkshake.
If Dimitri offers to pay the bill, how much will he have to pay?
1.2 Mr Zuma, Principal of the Perseverance Primary School, decides to replace the old floor tiles in the hall with new ones. The tiles are delivered and he asks the learners to carry them into the hall.
The Grade 7’s carry 1 893 tiles; the Grade 6’s carry 976 tiles, the Grade 5’s 824 tiles and the Grade 4’s 769 tiles to the hall. How many tiles have been carried to the hall?
2. Explain your group’s method of calculation to the class.
3. Compare your methods. How do they differ?
4. Use a pocket calculator to verify your answers.
Assessment
Learning Outcome 1: The 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 Standard 1.6: We know this when the learner solves problems in context including contexts that may be used to build awareness of other Learning Areas, as well as human rights,
social, economic and environmental issues;
Assessment Standard 1.10: We know this when the learner uses a range of techniques to perform written and mental calculations with whole numbers;
Assessment Standard 1.11: We know this when the learner uses a range of strategies to check solutions and judge the reasonableness of solutions.
1.19. To determine the equivalence and validity of
different representations*
MATHEMATICS
Number Concept, Addition and Subtraction
Addition
EDUCATOR SECTION
Memorandum
1.1 Counting on
1.2 Take away 11 400 every time and add the rest
1.3 Keep 1 000s, 100s and 10s apart and add
1.4 Counting on: first 10 000s, then 1 000s, then 100s, then 10s and lastly units
1.5 Self-explanatory
1.6 Carry over method
2. Own answer
3.2 Short method of addition, without carry over numbers
3.3
a) 83 320
b) 105 935
Leaner Section
Content
Activity: To determine the equivalence and validity of different
representations [LO 2.6]
To use strategies to check solutions [LO 1.11]
1. In Activity 2.6 you used your own techniques and strategies to solve the problems.
DID YOU KNOW?
There are many other ways in which we can add numbers. Form six groups. Each group must
discuss one of the following methods and then explain how the answer is calculated.
1.1 I calculate 11 468 + 23 957 like this:
11 468 + 20 000 → 31 468 + 3 000 → 34 468 + 900
35 368 + 50 → 35 418 + 7 = 35 425
1.2 I prefer to calculate the answer as follows:
11 468 + 23 957
= (11 400 + 11 400) + (68 + 12 557)
= (11 400 + 11 400 + 11 400) + (68 + 1 157)
= 34 200 + 1 225
= 35 425
1.3 Look carefully! This is how I do it!
