Mathematics Grade 5
Collection edited by: Siyavula Uploader
Content authors: Siyavula Uploader and Siyavula Uploader
Online: < http://cnx.org/content/col10994/1.3> This selection and arrangement of content as a collection is copyrighted by Siyavula Uploader.
It is licensed under the Creative Commons Attribution License: http://creativecommons.org/licenses/by/3.0/
Collection structure revised: 2009/09/23
For copyright and attribution information for the modules contained in this collection, see the " Attributions" section at the end of the collection.
Mathematics Grade 5
Table of Contents
1.1. To be able to do mental arithmetic
Number Concept, Addition and Subtraction
Activity: To be able to do mental arithmetic [LO 1.9]
1.2. To be able to solve problems in context
Number Concept, Addition and Subtraction
Activity: To be able to solve problems in context [LO 1.6]
To use a series of techniques to do calculations [LO 1.10]
1.3. To determine the equivalence and validity of different representations
Number Concept, Addition and Subtraction
Activity: To determine the equivalence and validity of different
To use strategies to check solutions [LO 1.11]
1.4. To use a series of techniques to do calculations
Number Concept, Addition and Subtraction
Activity: To be able to solve problems in context [LO 1.10]
To use strategies to check solutions [LO 1.11]
1.5. To solve problems in context
Number Concept, Addition and Subtraction
Activity: To solve problems in context [LO 1.6]
1.6. To be able to do mental arithmetic
Number Concept, Addition and Subtraction
ACTIVITY: To be able to do mental arithmetic [LO 1.9]
1.7. To count and calculate correctly without using a pencil and paper
Number Concept, Addition and Subtraction
ACTIVITY: To count and calculate correctly without using a pencil and paper
1.8. To recognise numbers and represent them correctly
Number Concept, Addition and Subtraction
Activity: To recognise numbers and represent them correctly [LO 1.3]
To recognise the place value of digits [LO 1.4]
1.9. To improve your mental arithmetic skills
Number Concept, Addition and Subtraction
ACTIVITY: To improve your mental arithmetic skills [LO 1.9]
To examine and extend numeric patterns [LO 2.1]
1.10. To recognise numbers and compare them to one another
Number Concept, Addition and Subtraction
Activity: To recognise numbers and compare them to one another [LO 1.3]
To be able to calculate correctly [LO 1.8]
1.11. To represent, recognise and compare numbers
Number Concept, Addition and Subtraction
Activity: To represent, recognise and compare numbers [LO 1.3]
To be able to calculate correctly [LO 1.8]
To improve mental arithmetic skills [LO 1.9]
TIME FOR SELF-ASSESSMENT ONCE MORE
1.12. To count forward in intervals
Number Concept, Addition and Subtraction
Activity: To count forward in intervals [LO 1.1]
To recognise, represent, describe and compare numbers, counting down and up
1.13. To be able to count correctly
Number Concept, Addition and Subtraction
Activity: To be able to count correctly [LO 1.1]
To be able to calculate correctly [LO 1.8]
To be able to do mental arithmetic [LO 1.9]
1.14. To use a series of techniques to do calculations
Number Concept, Addition and Subtraction
Activity: To use a series of techniques to do calculations [LO 1.10]
1.15. To describe and illustrate different cultures' ways of writing
Number Concept, Addition and Subtraction
Activity: To describe and illustrate different cultures' ways of writing [LO 1.2]
1.16. To recognise, describe and use laws
Number Concept, Addition and Subtraction
Activity: To recognise, describe and use laws [LO 1.12]
1.17. To estimate and calculate by selecting and using suitable operations
Number Concept, Addition and Subtraction
Activity: To estimate and calculate by selecting and using suitable operations
1.18. To solve problems in context
Number Concept, Addition and Subtraction
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]
1.19. To determine the equivalence and validity of different representations
Number Concept, Addition and Subtraction
Activity: To determine the equivalence and validity of different representations
To use strategies to check solutions [LO 1.11]
1.20. To use a series of techniques to do calculations
Number Concept, Addition and Subtraction
Activity: To use a series of techniques to do calculations [LO 1.10]
1.21. To be able to do mental arithmetic
Number Concept, Addition and Subtraction
Activity: To be able to do mental arithmetic [LO 1.9]
1.22. To use techniques to do calculations
Number Concept, Addition and Subtraction
Activity: To use techniques to do calculations [LO 1.10]
1.23. To solve in context problems that concern economic issues
Number Concept, Addition and Subtraction
Activity: To solve in context problems that concern economic issues [LO 1.6]
1.24. To be able to do mental arithmetic
Number Concept, Addition and Subtraction
Activity: To be able to do mental arithmetic [LO 1.9]
1.25. To describe and illustrate different cultures' ways of writing
Number Concept, Addition and Subtraction
Activity: To describe and illustrate different cultures' ways of writing [LO 1.2]
1.26. To use a series of techniques to be able to do calculations
Number Concept, Addition and Subtraction
Activity: To use a series of techniques to be able to do calculations [LO 1.10]
1.27. To examine and expand numeric patterns
Number Concept, Addition and Subtraction
Activity: To examine and expand numeric patterns [LO 2.1]
1.28. To determine the output values for given input values
Number Concept, Addition and Subtraction
Activity: To determine the output values for given input values [LO 2.3]
1.29. To calculate by using selected computations
Number Concept, Addition and Subtraction
Activity: To calculate by using selected computations [LO 1.8]
To recognise and represent the multiples of single-digit numbers [LO 1.3.6]
To recognise and represent the factors of at least any two-digit whole number
To be able to do mental arithmetic [LO 1.9.2]
To describe and illustrate different ways of writing in different cultures [LO
To use a series of techniques to do computations with whole numbers [LO
To calculate by using selected computations [LO 1.9.2]
To recognise and use the characteristics of multiplication with whole numbers
To estimate and calculate by means of rounding off [LO 1.8.1]
To use a series of techniques to do calculations [LO 1.10.5]
To be able to do mental arithmetic [LO 1.9.2]
To be able to solve problems in context [LO 1.6.1]
To be able to use a series of techniques to do calculations [LO 1.10.5]
To determine the equivalence and the validity of different representations [LO
To use strategies to check solutions [LO 1.11]
To solve problems in context [LO 1.6.1]
To be able to do mental arithmetic [LO 1.10]
To recognise, describe and use the reciprocal relation between multiplication
To recognise and represent numbers and to describe and compare them [LO
To describe and illustrate other cultures' ways of writing [LO 1.2]
To describe observed patterns and rules in own terms [LO 2.2]
To calculate by using selected computations [LO 1.8.4]
To be able to do mental arithmetic [LO 1.9.2]
To estimate and calculate by using selected computations [LO 1.8.5]
To be able to solve problems in context [LO 1.6.1]
To solve problems in context [LO 1.6]
To determine the equivalence and validity of different representations of the
same problem by means of comparison and discussion [LO 2.6]
To calculate by using selected computations [LO 1.8.5]
3.1. To recognise, classify and represent fractions (positive numbers)
ORDINARY AND DECIMAL FRACTIONS
TO RECOGNISE, CLASSIFY AND REPRESENT FRACTIONS (POSITIVE
To recognise, classify and represent fractions (positive numbers) in order to
describe and compare them [LO 1.3.2]
To recognise, classify and represent fractions (positive numbers) in order to
describe and compare them [LO 1.3.2]
To use tables to sort and record data [LO 5.3]
To calculate by means of computations that are suitable to be used in adding
To recognise and use equivalent forms [LO 1.5.1]
3.2. Recognise and classify ordinary fractions
ORDINARY AND DECIMAL FRACTIONS
RECOGNISE AND CLASSIFY ORDINARY FRACTIONS
To recognise and classify ordinary fractions in order to compare them [LO
To calculate by selecting and using operations [LO 1.8.3]
To simplify common fractions [LO 1.3.2]
To use a series of techniques to do calculations [LO 1.10.3]
To calculate through selection and use of suitable computations [LO 1.8.3]
To calculate through selection and the use of suitable computations [LO 1.8.3]
To recognise and use equivalent forms of fractions [LO 1.5.1]
To write number sentences in order to describe a problem situation [LO 2.4]
3.3. Selection and computations
ORDINARY AND DECIMAL FRACTIONS
To calculate through selection and by using suitable computations [LO 1.8.3]
To describe observed relationships and rules in your own words [LO 2.2]
To calculate through selection and by using suitable computations [LO 1.8.6]
To determine, through comparison and discussion, the equivalence and
validity of different representations of the same problem [LO 2.6.2]
To describe observed relationships and rules in your own words [LO 2.2]
To calculate through selection and by using suitable computations [LO 1.8.6]
To solve problems in context [LO 1.6.1]
To use a series of techniques to do calculations [LO 1.10.5]
3.4. Recognise, classify, represent and describe numbers
ORDINARY AND DECIMAL FRACTIONS
RECOGNISE, CLASSIFY, REPRESENT AND COMPARE NUMBERS
To recognise, classify, represent and compare numbers [LO 1.3.3]
To recognise and use equivalent forms of numbers [LO 1.5.2]
To use a series of techniques to do mental arithmetic [LO 1.10.5]
To be capable of doing mental arithmetic [LO 1.9]
To recognise, classify and represent numbers in order to describe and compare
To recognise and use equivalent forms of numbers [LO 1.5.2]
3.5. Tables and checks to arrange and record data
ORDINARY AND DECIMAL FRACTIONS
TABLES AND CHECKS TO ARRANGE AND RECORD DATA
To use tables and checks to arrange and record data [LO 5.3]
To recognise and represent numbers [LO 1.3.3]
To use a series of techniques to do mental arithmetic [LO 1.10.2]
To recognise and classify numbers in order to compare them [LO 1.3.3]
To use tables and checks in order to arrange and record data [LO 5.3]
To recognise, classify and represent numbers in order to describe and compare
To recognise, classify and represent numbers in order to describe and compare
3.6. Solve problems in context
ORDINARY FRACTIONS AND DECIMAL FRACTIONS
To solve problems in context [LO 1.6.2]
To determine the equivalence and validity of different representations of the
same problem through comparison and discussion [LO 2.6.3]
To calculate by means of selection and through the use of suitable
computations (additional) [LO 1.8]
To solve problems in context [LO 1.6.2]
To use a series of strategies to check solutions and to assess the
reasonableness of the solutions [LO 1.11]
To calculate through selection and the use of suitable computations
To solve problems in context [LO 1.6.2]
To solve problems that include selecting, calculating with and converting
To use standard units to record [LO 4.5.3]
To solve problems that include the selection of standard units [LO 4.6]
To measure and record [LO 4.5.1]
To measure accurately using appropriate measuring instruments [LO 4.7.1]
To estimate, measure and record [LO 4.5.1]
To measure accurately using appropriate measuring instruments [LO 4.7.1]
To solve problems that include selecting, calculating with and converting
GEOMETRY, DATA HANDLING AND PROBABILITY
To visualise and name 2-D shapes [LO 3.1.2]
To compare 2-D shapes with reference to certain qualities [LO 3.2.2]
To draw and describe 2-D shapes in terms of symmetry [LO 3.3.1]
To examine and compare 3D objects [LO 3.3.1]
To recognise 3D objects [LO 3.1.1]
To make patterns from geometric shapes and to describe them in terms of
Chapter 1. Term 1
1.1. To be able to do mental arithmetic*
MATHEMATICS
Number Concept, Addition and Subtraction
Subtraction
EDUCATOR SECTION
Memorandum
1.1 24
1.2 34
1.3 200
1.4 30
1.5 111
1.6 8
1.7 180
1.8 298
1.9 35
1.10 995
1.11 42
1.12 72
1.13 132
1.14 8
1.15 6
1.16 7
1.17 72
1.18 9
1.19 318
1.20 342
Leaner Section
Content
Activity: To be able to do mental arithmetic [LO 1.9]
1. Before we look at more methods of subtraction, you have another chance to improve your
mental arithmetic skills. Complete the following as quickly as you can.
1.1 38 – 14 = _________________
1.2 68 – 34 = _________________
1.3 _________________ – 101 = 99
1.4 110 – _________________ = 80
1.5 _________________ – 32 = 79
1.6 17 + _________________ = 25
1.7 260 + _________________ = 440
1.8 310 – 5 – 7 = _________________
1.9 400 – 15 – _________________ = 350
1.10 1 004 – 9 = _________________
1.11 6 × 7 = _________________
1.12 8 × 9 = _________________
1.13 12 × 11 = _________________
1.14 48 ÷ 6 = _________________
1.15 54 ÷ 9 = _________________
1.16 _________________ × 7 = 49
1.17 _________________ ÷ 6 = 12
1.18 81 ÷ _________________ = 9
1.19 Double 159: _________________
1.20 Halve 684: _________________
Colour in the appropriate block:
My mental calculation marks have IMPROVED
Table 1.1.
My mental calculation marks have IMPROVED
DETERIORATED
REMAINED THE SAME
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 1.9: We know this when the learner performs mental calculations.
1.2. To be able to solve problems in context*
MATHEMATICS
Number Concept, Addition and Subtraction
Subtraction
EDUCATOR SECTION
Memorandum
1.1 2 614
1.2 5 860
2-4 Own answer
Leaner Section
Content
Activity: To be able to solve problems in context [LO 1.6]
To use a series of techniques to do calculations [LO 1.10]
You have probably heard on the news or read in the papers that problems like the one described
below are very commonplace nowadays. We have to be able to do calculations correctly if we
want to find the answers to these problems!
1. Divide into groups of three. See whether you can solve the following problems. Your educator will supply you with the necessary paper to work on.
1.1 After terrible floods in their town, the Sunshine Primary School decides to contribute as many blankets as possible to flood victims. The Principal, Mrs Kuhn, says that about 10 000 blankets are needed. If the learners collect 7 386 blankets, how many more are needed?
1.2 Two factories produce sweets for disadvantaged children. If the Lollipop factory packaged 15
125 sweets and the Jelly Baby factory 20 985, how many fewer sweets did the Lollipop factory
have to distribute?
2. Explain to the rest of the class how your group calculated your answers.
3. Compare your methods. How do they differ?
4. Use a calculator to check the 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 such as:
1.6.1 financial (including buying and selling, profit and loss, and simple budgets);
Assessment Standard 1.10: We know this when the learner uses a range of techniques to perform written and mental calculations with whole numbers;
1.10.1 adding and subtracting in columns.
1.3. To determine the equivalence and validity of
different representations*
MATHEMATICS
Number Concept, Addition and Subtraction
Subtraction
EDUCATOR SECTION
Memorandum
2.1 and 2.2 own answer
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]
During the revious activity you used your own techniques and strategies to solve the problems.
In your feedback to the class you probably noticed that there is not only one way in which we
can subtract numbers. Divide into groups of three. Read the following problem thoroughly and
as a group work through the different methods of solving it.
A rugby match was attended by 32 564 men and 29 436 women.
How many more men than women watched the match?
1.1 I like doing addition:
32 564 – 29 436
Therefore: 29 436 + 64 = 29 500
29 500 + 500 = 30 000
30 000 + 2 564 = 32 564
64 + 500 + 2 564 = 3 128
Therefore there were 3 128 more men than women.
1.2 I round off the second number to the nearest 100:
32 564 – 29 436
Therefore: 32 564 – 29 400 = 3 164
3 164 – 36 = 3 128
The answer is 3 128 more men.
1.3 I prefer rounding off the subtrahend to the nearest 1 000:
32 564 – 29 436
Therefore: 32 564 – 29 000 = 3 564
3 564 – 436 = 3 128
1.4 I calculate the difference step by step:
32 564 – 29 436
Therefore: 32 000 – 29 000 = 3 000
564 – 436 = 128
3000 + 128 = 3 128
1.5 I first write the numbers in extended notation:
32 564 – 29 436
Thus: 30 000 + 2 000 + 500 + 60 + 4
- 20 000 + 9 000 + 400 + 30 + 6
Now I regroup:
20 000 + 12 000 + 500 + 50 + 14
- 20 000 + 9 000 + 400 + 30 + 6
0 + 3 000 + 100 + 20 + 8
Therefore the answer is 3 128
1.6 I calculate the difference by working with negative numbers:
32 564 – 29 436
Therefore: 30 000 – 20 000 = 10 000
2 000 – 9 000 = – 7 000 (I still have to subtract 7 000)
500 – 400 = 100
60 – 30 = 30
4 – 6 = – 2 (I still have to subtract 2)
The difference therefore is:
10 000 – 7 000 + 100 + 30 – 2 = 3 128
2. 2.1 Which of the above methods is the easiest for YOU? ______________
Why? _______________________________________________________________
_____________________________________________________________________
__________________________________________________________________________________________________________________________________________
2.2 See if your group can think of any another method for calculating the difference.
_____________________________________________________________________
__________________________________________________________________________________________________________________________________________
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. 11: We know this when the learner uses a range of strategies to check solutions and judge the reasonableness of solutions;
Learning Outcome 2: The learner will be able to recognise, describe and represent patterns and relationships, as well as to solve problems using algebraic language and skills.
Assessment Standard 2.6: We know this when the learner determines, through discussion and comparison, the equivalence of different descriptions of the same relationship or rule.
1.4. To use a series of techniques to do calculations*
MATHEMATICS
Number Concept, Addition and Subtraction
Subtraction
EDUCATOR SECTION
Memorandum
1.1 27 848
1.2 18 121
1.3 254 706
2.1 8 486 – 5 456 = 3 030
84 381 – 54 351 = 30 030
86 488 – 56 455 = 30 033
816 823 – 516 523 = 300 300
814 886 – 514 556 = 300 330
2.2 297 300
3.1 (b) R27 414
3.2 (a) 22 526
(b) 65 778
(c) 668 544
PUZZLE
(a) 378 – 111 = 267 – 110 = 157 – 10 = 147
(b) 2 598 – 1 111 = 1 487 – 1 111 = 376 – 110 = 266
(c) 14 210 – 110 = 14 100 – 100 = 14 000
Leaner Section
Content
Activity: To be able to solve problems in context [LO 1.10]
To use strategies to check solutions [LO 1.11]
1.1 __________ = 54 321 – 26 473
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
1.2 __________= 63 904 – 45 783
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
1.3 __________= 486 213 – 231 507
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
2. 2.1 Let's have a race! See if you can beat your partner, but remember to work neatly!
Take a good look at each number.
Replace the figure 8 with a 5 in each instance.
Write the new number right below the earlier one.
Then calculate the difference between the 2 numbers and write the answer in the last space.
e.g.
Figure 1.1.
Figure 1.2.
2.2 What is the difference between the biggest and the smallest answer?
_____________________________________________________________________
2.3 Compare your answers with those of your partner.
2.4 Check your answers with the help of a pocket calculator.
3. What follows is still another technique to calculate the difference between two numbers.
3.1 Work in groups of three again. Read the following problems attentively and explain the
solutions to one another.
a) Kayla's father buys a brand new Renault Scenic for R147 293 and Wayne's father buys a
second-hand car for R69 475. How much more does Kayla’s father pay than Wayne's father?
Figure 1.3.
Kayla's father pays R77 818 more.
b) Roberto frequently has to make overseas phone calls and his telephone account comes to R30
000. Carli uses her cell phone to make telephone calls and her account comes to R2 586. By what amount is Carli's account less than Roberto's account?
Figure 1.4.
Carli pays R27 414 less.
3.2 Calculate the following by making use of the above method:
a) __________= 42 063 – 19 537
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
b) __________= 264 136 – 198 358
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
c) __________= 900 000 – 231 456
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
TIME FOR SELF-ASSESSMENT
We have now studied different techniques / strategies to do and to check calculations. How do
YOU feel about the work we have just completed?
Assess yourself by circling the figure in the applicable column.
Table 1.2.
COMPLETELY
SLIGHTLY
FAIRLY
VERY
**
UNCERTAIN
CERTAIN
CERTAIN
CERTAIN
I can round off correctly to the
1
2
3
4
nearest 1 000
I can subtract by adding
1
2
3
4
I can estimate the difference by
1
2
3
4
means of rounding off
I can subtract by rearranging
1
2
3
4
numbers in extended notation
I can work with negative numbers
1
2
3
4
I can subtract vertically
1
2
3
4
Puzzle!
You may only use the following keys of your calculator:
1 ; – ; 0 ; =
See if you are able to change the first number into the second
number!
e.g. Change 1 416 to 285
1 416 – 1 110 = 306 – 11 = 295 – 10 = 285
a) Change 378 into 147
_____________________________________________________________________
b) Change 2 598 into 266
_____________________________________________________________________
c) Change 14 210 into 14 000
_____________________________________________________________________
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;
1.10.1 adding and subtracting in columns.
Assessment Standard 1.11: We know this when the learner uses a range of strategies to check solutions and judge the reasonableness of solutions.
1.5. To solve problems in context*
MATHEMATICS
Number Concept, Addition and Subtraction
Subtraction
EDUCATOR SECTION
Memorandum
Own opinion
TEST 3 TOTAL: 20
1. (a) Difference
1. Addition
2. Minuend
2. (a) 105; 70;
(b) 480; 360;
Figure 1.5.
4. 46 – 000 – 23 000 = 23 000
5. 1 450
6. 23 600 + 400 = 24 000
24 000 + 32 123 = 56 123
32 123 + 400 + 31 = 32 554
7.
Figure 1.6.
8. R500 000 – R13 401
R486 599
Leaner Section
Content
Activity: To solve problems in context [LO 1.6]
CHALLENGE
Here is an assignment for your portfolio. Your educator will go through it with you and explain exactly what is expected of you. Make sure that you understand completely before you begin.
Remember to work neatly!
A challenge!
You have won a competition and are richer by a million rand! But you may only buy articles
advertised in your local newspaper. Cut out the pictures/advertisements of what you would buy
and paste them below, together with the prices. Indicate how much money you will have left after making your purchases. Ask a friend to check your calculations.
ASSESSMENT: ACTIVITY 3.11
Table 1.3.
1
2
3
4
Neatness
Work is untidy Organised,
Neat, well
Neat and organised, easily
and
and un
but hardly
organised and
legible
organisation organised
legible
easily legible
Logical
Hardly any
A little logic
Uses effective Mathematical Uses refined logic
reasoning
proof of logic
present
reasoning
and reasoning
Correctness
Answers wrong A few errors
Sufficient understanding, but
of
Answer is correct
(items omitted) committed
subtraction errors committed
calculations
TEST
SEE HOW WELL YOU CAN COPE!
1. Write down the correct answer:
1.1 The answer of a subtraction sum is known as the _______________________