Intermediate Phase Grade 4 • Facilitator’s Guide
Mathematics CAPS IEB Mathematics
• Comprehensive explanations of concepts in simple language. everyday examples with visuals and diagrams to help • Practical, master concepts. • Learners work at their own pace. • Activities that test learners’ knowledge application and reasoning. • The facilitator’s guide contains step-by-step calculations and answers. • Use in school or at home.
home classroom college workplace
9 781990
Facilitator’s Guide
2004-E-MAM-FG01
946349
4
Mathematics Facilitator’s guide Grade 4
2004-E-MAM-FG01
9 781990
946349
CAPS aligned
L Young
Facilitator’s Guide G04 ~ Mathematics
Contents LESSON ELEMENTS.....................................................................................................................................1 INTRODUCTION...........................................................................................................................................2 TIME MANAGEMENT..................................................................................................................................2 ASSESSMENT REQUIREMENTS................................................................................................................7 YEAR PLAN....................................................................................................................................................8 UNIT 1.......................................................................................................................................................... 12 LESSON 1: WHOLE NUMBERS............................................................................................................... 13 Counting with whole numbers....................................................................................................................14 ACTIVITY 1...............................................................................................................................................16 Ordering whole numbers...............................................................................................................................19 ACTIVITY 2...............................................................................................................................................20 Place value............................................................................................................................................................22 ACTIVITY 3...............................................................................................................................................24 Comparing whole numbers...........................................................................................................................29 ACTIVITY 4...............................................................................................................................................30 Representing whole numbers......................................................................................................................33 ACTIVITY 5...............................................................................................................................................33 LESSON 2: NUMBER SENTENCES......................................................................................................... 34 ACTIVITY 6...............................................................................................................................................35 ACTIVITY 7...............................................................................................................................................36 ACTIVITY 8...............................................................................................................................................37 Pairs of equivalent number sentences.....................................................................................................39 ACTIVITY 9...............................................................................................................................................40 Addition and subtraction techniques.......................................................................................................41 Commutative property of addition............................................................................................................41 Associative property of addition................................................................................................................42 ACTIVITY 10.............................................................................................................................................43 Order of subtraction.........................................................................................................................................44 ACTIVITY 11.............................................................................................................................................45 LESSON 3: WHOLE NUMBERS............................................................................................................... 46 Addition with whole numbers.....................................................................................................................46 ACTIVITY 12.............................................................................................................................................46 ACTIVITY 13.............................................................................................................................................49 Subtraction of whole numbers....................................................................................................................51 ACTIVITY 14.............................................................................................................................................52 Estimation............................................................................................................................................................58 ACTIVITY 15.............................................................................................................................................60 Rounding...............................................................................................................................................................63 ACTIVITY 16.............................................................................................................................................65 ACTIVITY 17.............................................................................................................................................67 Compensation.....................................................................................................................................................68 Doubling and halving.......................................................................................................................................68 i
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Facilitator’s Guide G04 ~ Mathematics
ACTIVITY 18.............................................................................................................................................69 ACTIVITY 19.............................................................................................................................................71 Testing calculations with inverse operations........................................................................................72 ACTIVITY 20.............................................................................................................................................73 LESSON 4: NUMBER PATTERNS (Numeric patterns).................................................................... 75 ACTIVITY 21.............................................................................................................................................76 Input and output values.................................................................................................................................77 ACTIVITY 22.............................................................................................................................................77 ACTIVITY 23.............................................................................................................................................79 ACTIVITY 24.............................................................................................................................................80 The associative property of multiplication............................................................................................81 LESSON 5: WHOLE NUMBERS (Multiplication and division)..................................................... 84 Multiplication......................................................................................................................................................84 ACTIVITY 25.............................................................................................................................................92 Division..................................................................................................................................................................93 ACTIVITY 26.............................................................................................................................................94 ACTIVITY 27.............................................................................................................................................97
LESSON 6: TIME........................................................................................................................................ 98 ACTIVITY 28..........................................................................................................................................101 ACTIVITY 29..........................................................................................................................................104 LESSON 7: DATA HANDLING................................................................................................................106 Collecting, organising, analysing and interpreting data................................................................106 Pictograph.........................................................................................................................................................106 Pie chart..............................................................................................................................................................107 Bar graph...........................................................................................................................................................108 ACTIVITY 30..........................................................................................................................................109 LESSON 8: PROPERTIES OF 2D SHAPES..........................................................................................113 ACTIVITY 31..........................................................................................................................................116
LESSON 9: WHOLE NUMBERS (Multiplication of 2-digit numbers and division of 2-digit numbers by 1-digit numbers)...........................................................................................................118 ACTIVITY 32..........................................................................................................................................118 ACTIVITY 33..........................................................................................................................................120 ACTIVITY 34..........................................................................................................................................124 UNIT 2........................................................................................................................................................126 LESSON 10: WHOLE NUMBERS..........................................................................................................127 Counting with whole numbers.................................................................................................................128 ACTIVITY 35..........................................................................................................................................129 Ordering whole numbers............................................................................................................................132 ACTIVITY 36..........................................................................................................................................133 Place value.........................................................................................................................................................135 © Impaq
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Facilitator’s Guide G04 ~ Mathematics
ACTIVITY 37..........................................................................................................................................137 Comparing whole numbers........................................................................................................................140 ACTIVITY 38..........................................................................................................................................141 Representing whole numbers...................................................................................................................143 ACTIVITY 39..........................................................................................................................................143 Rounding............................................................................................................................................................143 ACTIVITY 40..........................................................................................................................................146 LESSON 11: WHOLE NUMBERS (Addition and subtraction)...................................................148 Commutative property of addition.........................................................................................................148 Associative property of addition.............................................................................................................149 Order of subtraction......................................................................................................................................151 ACTIVITY 41..........................................................................................................................................153 LESSON 12: COMMON FRACTIONS....................................................................................................156 ACTIVITY 42..........................................................................................................................................158 ACTIVITY 43..........................................................................................................................................160 LESSON 13: LENGTH..............................................................................................................................167 ACTIVITY 44..........................................................................................................................................168 LESSON 14: WHOLE NUMBERS (Multiply 2-digit numbers by 2-digit numbers)..............171 Distributive property of multiplication................................................................................................173 Dividing numbers into factors to multiply them..............................................................................173 ACTIVITY 45..........................................................................................................................................174 LESSON 15: PROPERTIES OF 3D OBJECTS......................................................................................179 ACTIVITY 46..........................................................................................................................................182 LESSON 16: GEOMETRIC PATTERNS................................................................................................185 ACTIVITY 47..........................................................................................................................................187 LESSON 17: SYMMETRY.......................................................................................................................189 ACTIVITY 48..........................................................................................................................................190 LESSON 18: WHOLE NUMBERS (Addition and subtraction)....................................................192 ACTIVITY 49..........................................................................................................................................192 LESSON 19: WHOLE NUMBERS (Dividing 3-digit numbers by 1-digit number)................196 ACTIVITY 50..........................................................................................................................................200 UNIT 3........................................................................................................................................................206 LESSON 20: CAPACITY/VOLUME.......................................................................................................207 ACTIVITY 51..........................................................................................................................................211 LESSON 21: COMMON FRACTIONS....................................................................................................216 ACTIVITY 52..........................................................................................................................................219 iii
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Facilitator’s Guide G04 ~ Mathematics
LESSON 22: WHOLE NUMBERS .........................................................................................................224 ACTIVITY 53..........................................................................................................................................225 LESSON 23: VIEWS OF OBJECTS.........................................................................................................232 ACTIVITY 54..........................................................................................................................................233 LESSON 24: PROPERTIES OF 2D SHAPES........................................................................................237 ACTIVITY 55..........................................................................................................................................240 LESSON 25: DATA HANDLING.............................................................................................................244 ACTIVITY 56..........................................................................................................................................246 LESSON 26: NUMBER PATTERNS.......................................................................................................251 Input and output values..............................................................................................................................252 The associative property of multiplication.........................................................................................254 Types of number sequences......................................................................................................................256 ACTIVITY 57..........................................................................................................................................257 LESSON 27: WHOLE NUMBERS (Addition and subtraction [4-digit whole numbers])...260 Order of subtraction......................................................................................................................................262 ACTIVITY 58..........................................................................................................................................264
LESSON 28: WHOLE NUMBERS (Multiply [2-digit whole numbers by 2-digit whole numbers])................................................................................................................................................269 Distributive property of multiplication................................................................................................269 Dividing numbers into factors to multiply them..............................................................................270 ACTIVITY 59..........................................................................................................................................271 LESSON 29: NUMBER SENTENCES....................................................................................................276 Pairs of equivalent number sentences..................................................................................................279 ACTIVITY 60..........................................................................................................................................280 LESSON 30: TRANSFORMATIONS......................................................................................................283 ACTIVITY 61..........................................................................................................................................284 UNIT 4........................................................................................................................................................285 LESSON 31: WHOLE NUMBERS..........................................................................................................286 ACTIVITY 62..........................................................................................................................................286 LESSON 32: MASS...................................................................................................................................295 ACTIVITY 63..........................................................................................................................................298 LESSON 33: PROPERTIES OF 3D OBJECTS......................................................................................304 ACTIVITY 64..........................................................................................................................................306 LESSON 34: COMMON FRACTIONS....................................................................................................310
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Facilitator’s Guide G04 ~ Mathematics
ACTIVITY 65..........................................................................................................................................313 LESSON 35: WHOLE NUMBERS (Division [3-digit numbers by 1-digit number]).............319 ACTIVITY 66..........................................................................................................................................321 LESSON 36: PERIMETER, SURFACE AREA AND VOLUME...........................................................325 ACTIVITY 67..........................................................................................................................................331 Surface area......................................................................................................................................................334 ACTIVITY 68..........................................................................................................................................335 Volume................................................................................................................................................................336 LESSON 37: POSITION AND MOVEMENT........................................................................................339 ACTIVITY 69..........................................................................................................................................341 LESSON 38: TRANSFORMATIONS......................................................................................................344 ACTIVITY 70..........................................................................................................................................346 LESSON 39: GEOMETRIC PATTERNS................................................................................................348 ACTIVITY 71..........................................................................................................................................350 LESSON 40: WHOLE NUMBERS (Addition and subtraction [4-digit whole numbers])...353 ACTIVITY 72..........................................................................................................................................353 LESSON 41: PROBABILITY...................................................................................................................359 ACTIVITY 73..........................................................................................................................................360 REFERENCES: UNIT 1............................................................................................................................362 REFERENCES: UNIT 2............................................................................................................................363 REFERENCES: UNIT 3............................................................................................................................364 REFERENCES: UNIT 4............................................................................................................................365 100 CHART...............................................................................................................................................366 ADDENDUM A..........................................................................................................................................367
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Facilitator’s Guide G04 ~ Mathematics
LESSON ELEMENTS The guide consists of various lesson elements. Every element is important for the learning process and indicates the skill that the learner needs to master. ICON
LESSON ELEMENT
Think for yourself
Tips
Research
Study New concept or definition Remember/Revise Take note! Important
Self-assessment Activity
1
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Facilitator’s Guide G04 ~ Mathematics
INTRODUCTION Grade 4 learners are entering a new phase in their school careers and are being introduced to the concept of self-study for the first time. The study guide and the facilitator’s guide are written to guide learners and facilitators through the process and lay the foundation for future academic success. It is important to remember that the facilitator’s support is still crucial to help learners gain self-confidence to master the new challenges. The study guide is written in a friendly and informal manner to involve learners and make the subject accessible and interesting. It consists of theory, activities and research activities. All activities in the study guide must be completed to help learners understand and apply the new knowledge.
After each lesson, learners must complete an individual self-assessment. Use the assessment to determine whether learners still need help with a particular lesson, and address the gap immediately. It may also be used to plan enrichment activities. Once learners have mastered the concepts in each lesson, additional enrichment activities may be completed. The facilitator’s guide guides the facilitator to support the learners so that the theory of mathematics is well established at the end of each lesson. Many of the lessons have a practical component which the facilitator must present. Use the guidelines in the facilitator’s guide. The study guide and facilitator’s guides are divided into four units. You are encouraged to complete one unit per term.
Do 10 minutes of mental arithmetic every day. Use the supplementary Train Your Brain Maths Grade 4 product. The CAMI exercise programme (www.camiweb.com) is recommended for revision and additional practice. This product consists of: • A facilitator’s guide • Two study guides: Units 1 and 2 are in study guide 1/2 and units 3 and 4 are in study guide 2/2. • Train Your Brain Maths Grade 4 (for mental arithmetic)
TIME MANAGEMENT
The time allocation per topic is merely a guideline and may be adapted according to the learners’ pace. It is important, however, to know which lessons must be completed before any test or exam can be attempted.
Learners must spend six hours per week on mathematics. Take note that this does not include all activities, assessments and examinations. If learners are working at a pace slower than the allotted time, make the necessary adjustments so that they can still complete all the work in time.
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Facilitator’s Guide G04 ~ Mathematics
It is also crucial that the learners do not progress to the next topic before they have a thorough understanding of the current topic. It is important to do revision until they have fully mastered the new concepts. Grade 4 is the start of a new phase and the last opportunity for learners to master the basics of mathematics. Guidelines for time allocation per topic UNIT 1 TOPIC
TIME AND NOTES
Mental maths: Use the Train Your Brain Maths Grade 4 product
LESSON 1 Whole numbers: Counting, ordering, comparing and representing, and place value of digits (3-digit whole numbers) LESSON 2 Number sentences
LESSON 3 Whole numbers: Addition and subtraction (3-digit whole numbers) LESSON 4 Number patterns: Numeric patterns
LESSON 5 Whole numbers: Multiplication and division (1-digit whole number by 1-digit whole number) LESSON 6 Time
8 hours (divided into 10 minutes every day) 2 hours 3 hours 8 hours 4 hours 4 hours 6 hours
LESSON 7 Data handling
10 hours
LESSON 9 Whole numbers: • Multiply (2-digit whole numbers by 1-digit whole number) • Multiply (2-digit whole numbers by 2-digit whole numbers) • Divide (2-digit whole numbers by 1-digit whole number)
5 hours
LESSON 8 Properties of 2D shapes
Revision: Use the CAMI programme TOTAL
3
5 hours
5 hours
60 hours
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Facilitator’s Guide G04 ~ Mathematics
UNIT 2 TOPIC
TIME AND NOTES
Mental maths: Use the Train Your Brain Maths Grade 4 product
LESSON 10 Whole numbers: Counting, ordering, comparing and representing, and place value of digits (4-digit whole numbers)
7 hours (divided into 10 minutes every day) 1 hour
LESSON 11 Whole numbers: Addition and subtraction (4-digit whole numbers)
4 hours
LESSON 13 Length
7 hours
LESSON 12 Common fractions
LESSON 14 Whole numbers: Multiplication (2-digit whole numbers by 2-digit whole numbers) LESSON 15 Properties of 3D shapes LESSON 16 Geometric patterns
6 hours 5 hours 4 hours
LESSON 17 Symmetry
LESSON 18 Whole numbers: Addition and subtraction (4-digit whole numbers) LESSON 19 Whole numbers: Division (3-digit whole numbers by 1-digit whole number) Revision: Use the CAMI programme TOTAL
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6 hours
2 hours 4 hours 4 hours 4 hours
60 hours
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Facilitator’s Guide G04 ~ Mathematics
UNIT 3 TOPIC
TIME AND NOTES
Mental maths: Use the Train Your Brain Maths Grade 4 product LESSON 20 Capacity/Volume
LESSON 21 Common fractions
LESSON 22 Whole numbers: Counting, ordering, comparing and representing, and place value of digits (4-digit whole numbers) Whole numbers: Addition and subtraction (4-digit whole numbers)
8 hours (divided into 10 minutes every day) 6 hours 5 hours 5 hours
LESSON 23 Views of objects
2 hours
LESSON 25 Data handling
7 hours
LESSON 24 Properties of 2D shapes
4 hours
LESSON 26 Numerical patterns
LESSON 27 Whole numbers: Addition and subtraction (4-digit whole numbers)
LESSON 28 Whole numbers: Multiplication (2-digit whole numbers with 2-digit whole numbers) LESSON 29 Number sentences LESSON 30 Transformations
4 hours 4 hours 5 hours 3 hours 3 hours
Revision: Use the CAMI programme
4 hours
TOTAL
60 hours
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Facilitator’s Guide G04 ~ Mathematics
UNIT 4 TOPIC
TIME AND NOTES
Mental maths: Use the Train Your Brain Maths Grade 4 product
LESSON 31 Whole numbers: Counting, ordering, comparing and representing, and place value of digits (4-digit whole numbers) Whole numbers: Addition and subtraction (4-digit whole numbers)
7 hours (divided into 10 minutes every day) 5 hours
LESSON 32 Mass
6 hours
LESSON 34 Common fractions
5 hours
LESSON 33 Properties of 3D objects
4 hours
LESSON 35 Whole numbers: Division (3-digit whole numbers by 1-digit whole number) LESSON 36 Perimeter, surface area and volume LESSON 37 Position and movement
7 hours 2 hours
LESSON 38 Transformations
3 hours
LESSON 39 Geometric patterns
LESSON 40 Whole numbers: Addition and subtraction (4-digit whole numbers) LESSON 41 Probability
Revision: Use the CAMI programme
2 hours 3 hours 2 hours 4 hours
TOTAL
60 hours
Find unit 1 and 2 in study guide 1/2 and unit 3 and 4 in study guide 2/2.
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3 hours
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Facilitator’s Guide G04 ~ Mathematics
ASSESSMENT REQUIREMENTS Formal assessment tasks and tests form part of a year-long formal assessment programme.
Formal assessment tasks and tests count 75% of the final mark and the November examinations count 25%.
The formal assessment programme indicates which components must be completed in each term (this includes the activities and investigations in the study guide). Term (Unit)
Formal assessment
1
• •
1 × assignment 1 × test
2
3
• •
1 × test 1 × project
2
• •
2
4
TOTAL • •
Number of formal assessments per term/unit
• • •
1 × test 1 × June examination 1 × assignment 1 × research 1 × November examination
Total % contribution to final mark
2
75
3
25
9
100
The June examination covers all the work in term 1 and 2 (unit 1 and 2). The November examination covers all the work in term 3 and 4 (unit 3 and 4).
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Facilitator’s Guide G04 ~ Mathematics
YEAR PLAN UNIT 1 TOPIC
Mental maths: Use the Train Your Brain Maths Grade 4 product
LESSON 1 Whole numbers: Counting, ordering, comparing and representing, and place value of digits (3-digit whole numbers) LESSON 2 Number sentences
LESSON 3 Whole numbers: Addition and subtraction (3-digit whole numbers) LESSON 4 Number patterns: Numeric patterns
LESSON 5 Whole numbers: Multiplication and division (1-digit whole number by 1-digit whole number) LESSON 6 Time
LESSON 7 Data handling
LESSON 8 Properties of 2D shapes
LESSON 9 Whole numbers: • Multiply (2-digit whole numbers by 1-digit whole number) • Multiply (2-digit whole numbers by 2-digit whole numbers) • Divide (2-digit whole numbers by 1-digit whole number) Revision: Use the CAMI programme © Impaq
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DATE
DATE
STARTED
COMPLETED
Facilitator’s Guide G04 ~ Mathematics
UNIT 2 TOPIC
DATE
DATE
STARTED
COMPLETED
Mental maths: Use the Train Your Brain Maths Grade 4 product LESSON 10 Whole numbers: Counting, ordering, comparing and representing, and place value of digits (4-digit whole numbers) LESSON 11 Whole numbers: Addition and subtraction (4-digit whole numbers) LESSON 12 Common fractions LESSON 13 Length
LESSON 14 Whole numbers: Multiplication (2-digit whole numbers by 2-digit whole numbers) LESSON 15 Properties of 3D shapes LESSON 16 Geometric patterns LESSON 17 Symmetry LESSON 18 Whole numbers: Addition and subtraction (4-digit whole numbers) LESSON 19 Whole numbers: Division (3-digit whole numbers by 1-digit whole number) Revision: Use the CAMI programme 9
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Facilitator’s Guide G04 ~ Mathematics
UNIT 3 TOPIC
Mental maths: Use the Train Your Brain Maths Grade 4 product LESSON 20 Capacity/Volume
LESSON 21 Common fractions
LESSON 22 Whole numbers: Counting, ordering, comparing and representing, and place value of digits (4-digit whole numbers) Whole numbers: Addition and subtraction (4-digit whole numbers) LESSON 23 Views of objects
LESSON 24 Properties of 2D shapes LESSON 25 Data handling
LESSON 26 Numerical patterns
LESSON 27 Whole numbers: Addition and subtraction (4-digit whole numbers) LESSON 28 Whole numbers: Multiplication (2-digit whole numbers with 2-digit whole numbers) LESSON 29 Number sentences LESSON 30 Transformations
Revision: Use the CAMI programme © Impaq
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DATE
DATE
STARTED
COMPLETED
Facilitator’s Guide G04 ~ Mathematics
UNIT 4 TOPIC
DATE
DATE
STARTED
COMPLETED
Mental maths: Use the Train Your Brain Maths Grade 4 product
LESSON 31 Whole numbers: Counting, ordering, comparing and representing, and place value of digits (4-digit whole numbers) Whole numbers: Addition and subtraction (4-digit whole numbers) LESSON 32 Mass
LESSON 33 Properties of 3D objects LESSON 34 Common fractions
LESSON 35 Whole numbers: Division (3-digit whole numbers by 1-digit whole number) LESSON 36 Perimeter, surface area and volume LESSON 37 Position and movement LESSON 38 Transformations
LESSON 39 Geometric patterns
LESSON 40 Whole numbers: Addition and subtraction (4-digit whole numbers) LESSON 41 Probability
Revision: Use the CAMI programme 11
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Unit
1
Facilitator’s Guide G04 ~ Mathematics
UNIT 1 This unit covers nine lessons: Lessons 1 to 9. The table indicates the lesson topics as well as the time you will spend on each one. UNIT 1
TOPIC
TIME AND NOTES
Mental maths: Use the Train Your Brain Maths Grade 4 product
LESSON 1 Whole numbers: Counting, ordering, comparing and representing, and place value of digits (3-digit whole numbers) LESSON 2 Number sentences
LESSON 3 Whole numbers: Addition and subtraction (3-digit whole numbers) LESSON 4 Number patterns: Numeric patterns
LESSON 5 Whole numbers: Multiplication and division (1-digit whole number by 1-digit whole number) LESSON 6 Time
LESSON 7 Data handling
2 hours 3 hours 8 hours 4 hours 4 hours 6 hours 10 hours
LESSON 8 Properties of 2D shapes
LESSON 9 Whole numbers: • Multiply (2-digit whole numbers by 1-digit whole number) • Multiply (2-digit whole numbers by 2-digit whole numbers) • Divide (2-digit whole numbers by 1-digit whole number) Revision: Use the CAMI programme TOTAL © Impaq
8 hours (divided into 10 minutes every day)
5 hours
5 hours 5 hours
60 hours 12
Facilitator’s Guide G04 ~ Mathematics
Unit
1
LESSON 1: WHOLE NUMBERS In this lesson learners will work with different notations of numbers for the first time. It is important to ensure that learners understand notations. It lays the foundation for the language of mathematics
and if learners are already comfortable with this, it can simplify the introduction of new mathematical
concepts. In this lesson, learners will work with whole numbers and do the following: • count • order • compare • represent • indicate place value (which is very important for the calculations that will be done later in the term) Do you know what a whole number is? Whole numbers are numbers without fractions or decimals. Whole numbers are always positive and never negative. Remember: 0 is also a whole number.
Learners were introduced to whole numbers in Grade 3, even if they did not necessarily call them whole numbers. Ensure that learners know that a whole number is not a fraction. Different games may be used to reinforce this concept.
Example Write down different numbers, including fractions and whole numbers. Learners close their eyes and randomly point to a place on the page. They identify the number they chose as a whole number or not. Examples of whole numbers {0; 1; 2; 3; 4; 5; 6; ...}
If numbers such as the above are placed in curly brackets { }, we call it a set of numbers. This means that {0; 1; 2; 3; 4; 5; 6; ...} is a set of whole numbers.
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Unit
1
Facilitator’s Guide G04 ~ Mathematics
Study the numbers in the table and circle the whole numbers.
-
15
5
0,9
101
10
24
-24
0
3
-4,1
-2
13 9
7
11 198 4
In this lesson, you will do the following with whole numbers: • count • order • compare • represent • indicate place value
Counting with whole numbers
It is very important that the learners count every day – both on (forwards) and back (backwards). In the beginning, learners may count with a 100 chart, but when they have completed the lesson, they must be able to count without any help. (See the 100 chart at the back of this facilitator’s guide.)
Counting must not only be seen as oral counting – learners must understand the calculations that are happening to count on or back.
Learners may use the following to count: •
countable objects (marbles, balls, beans, beads, etc.)
•
rows or diagrams, for example:
• •
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an abacus (available at any educational or plastics shop) number charts or tables
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Facilitator’s Guide G04 ~ Mathematics
Unit
1
In Grade 3 you learnt how to count with whole numbers. Do you still remember how to count in 2s? Let’s revise counting in 2s.
When you count in 2s, always add 2 to the previous number to get the next number.
+2
2
4
+2
6
+2
+2
8
10
+2
12
You can start with any number. Study the numbers. Do you see that you can start at any number and count in whole numbers?
+2
63
65
+2
67
+2
+2
69
71
+2
73
In the above examples, you counted on or forwards.
Using whole numbers, you can also count back or backwards.
Study the following numbers and complete the missing numbers. Count back in 3s.
–3
53
–3
50
–3
47
–3
–3
41
44
–3
38
–3
35
32
Now that your memory has been refreshed, you can apply what you have learnt.
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Unit
1
Facilitator’s Guide G04 ~ Mathematics
ACTIVITY 1 1. 2.
Write the set of whole numbers between 915 and 921. {916; 917; 918; 919; 920}
Indicate whether the numbers are whole numbers. Colour the correct circle. Numbers
Whole number: yes or no?
658 14,3 53 3 1 412 3.
Study the number lines. You are not told what number to count with, nor whether you are counting on or back. Complete the number lines. 96;
101;
106;
111
...
3.2
416
413;
410;
407;
...
8;
108;
208
308;
...
3.4
12;
9;
6
3;
0
10;
35;
60;
85;
110
3.1
3.3
3.5
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Facilitator’s Guide G04 ~ Mathematics
4.
1
Unit
Andrew, Sibongile and Timmy like to play with marbles. They keep their marbles in bags.
Andrew’s bag can hold 10 marbles. Sibongile’s bag can hold 25 marbles. Timmy’s bag can only hold 3 marbles. Look at the following example and count in 10s, 25s and 3s to determine how many marbles each child has. Child
5.
Number of bags
Number of marbles
Andrew
30
Sibongile
25
Timmy
27
Complete the flow charts. 5.1
+ 50
81
+ 50
131
+ 50
181 + 50
31
231 5.2
– 25
387
– 25
362
– 25
337 – 25
412
312
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Unit
1
Facilitator’s Guide G04 ~ Mathematics
After each lesson learners must complete a self-assessment of the completed lesson. You may use the self-assessment to determine whether learners need additional help in the specific lesson, and to address the gap immediately. You may also use the self-assessment to plan enrichment activities. When learners have mastered each lesson, additional enrichment will still be hugely beneficial.
Self-assessment Do the learners understand the work? Let them colour the faces that show what they can do. COUNTING IN WHOLE NUMBERS Requirements
Can the learners do it?
I can count on and back in 2s. I can count on and back in 3s. I can count on and back in 5s. I can count on and back in 10s. I can count on and back in 25s. I can count on and back in 50s. I can count on and back in 100s. I can do all of the above up to the number 1 000.
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Facilitator’s Guide G04 ~ Mathematics
Unit
1
Ordering whole numbers Ordering means to arrange or organise numbers.
order = arrange
We can order or arrange numbers in different ways:
from GREAT to small OR
from small to GREAT
Study the number set.
{25; 2; 537; 119; 65} Arrange the numbers from great to small. Step 1:
Choose the greatest number and write it down first.
The greatest number in this set is Step 2:
537.
Now cross out the greatest number in the number set.
{25; 2; 537; 119; 65}
You can no longer choose the number 537.
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Unit
Step 3:
1
Facilitator’s Guide G04 ~ Mathematics
From the remaining numbers, choose the greatest number and write it next to 537.
537; 119
Step 4:
Repeat steps 2 and 3 until you have crossed out all of the numbers.
When you are done, the set of numbers must look as follows:
{537; 119; 65; 25; 2} You have arranged the number set from greatest to smallest. Can you arrange the following number set from small to great?
{913; 902; 35; 11; 1} Tip: Start by choosing the smallest number, not the greatest one. Write your answer in the box below.
1; 11; 35; 902; 913
ACTIVITY 2 1.
Arrange the numbers from small to great. Question
1.1
321 645 256 782 127
127; 256; 321; 645; 782
1.4
65 34 12 47 71
12; 34; 47; 65; 71
1.2 1.3 2.
Answer
1.5
56 72 201 114 98
56; 72; 98; 114; 201
527 471 298 638 138
138; 298; 471; 527; 638
147; 274; 355; 568; 724
724 568 147 274 355
Arrange the numbers from great to small. Question
2.1
620 541 635 514 623
635; 623; 620; 541; 514
2.4
58 39 44 28 13
58; 44; 39; 28; 13
2.2 2.3 2.5 © Impaq
Answer
36 48 21 65 57
65; 57; 48; 36; 21
457 325 147 471 352
471; 457; 352; 325; 147
116; 110; 98; 76; 53
116 98 53 76 110
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Facilitator’s Guide G04 ~ Mathematics
3.
Build and arrange the numbers.
Question
3.1 3.2 3.3 3.4 3.5
1
Unit
Answer
Write down the numbers you can build with 1, 4 and 5 and arrange them from great to small. Write down the numbers you can build with 7, 2 and 9 and arrange them from small to great. Write down the numbers you can build with 8, 1 and 6 and arrange them from great to small. Write down the numbers you can build with 9, 1 and 2 and arrange them from small to great.
541; 514; 451; 415; 154; 145
279; 297; 729; 792; 927; 972
861; 816; 681; 618; 186; 168
129; 192; 219; 291; 912; 921
Write down the numbers you can build with 9, 1 and 1 and arrange them from great to small.
911; 191; 119
Self-assessment Do the learners understand the work? Let them colour the faces that show what they can do. ORDERING WHOLE NUMBERS Requirements
Can the learners do it?
I can arrange number sets from great to small. I can arrange number sets from small to great. I can build and arrange different numbers.
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Place value Place value is extremely important for developing number concept. If learners’ number concept is not
firmly established, they will find calculations difficult. Later in the term, calculations will already require a firm grasp of place value, which is why learners must spend enough time on this topic. Ensure that learners understand the concept really well.
Use additional resources to ensure the proper comprehension of place value, such as: • • •
100 charts
Flash cards
Dienes cubes
Various methods are covered in this topic. Learners must be aware of all the different methods, but they
are only required to use one method during assessment. The facilitator and learners together decide which method will work best.
Visit goo.gl/nBhGsR to find additional activities.
Place value helps us to determine the value of a digit.
Our number system (the numbers that we work with) contains digits from 0 to 9 only.
What do we do when we must work with numbers greater than 9?
We use place value to indicate when we are working with numbers greater than 9. It means that the value of a digit is determined by its place or position in a number.
When we work with place value, we can think of each value as a box with high walls.
Hundreds
Tens
Units
In each box you can find the digits 0 to 9. As soon as the number becomes greater than 9, it jumps over the wall into the next box. © Impaq
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Facilitator’s Guide G04 ~ Mathematics
Unit
Hundreds
Tens
1
Units 7
The 7 in the Units box means that there are 7 Units. We can also write it as 7 x 1. Can you see that there are no numbers in the Tens and Hundreds boxes? This means that there are 0 x Tens and 0 x Hundreds in this number. Let’s see what happens with a number greater than 9.
Hundreds
Tens
Units 13
There may only be one number in each box, but now there is a 1 and a 3 in the Units box. What does the 13 mean? It consists of 10 + 3.
The 3 is in the Units box and it means 3 x 1, which is shown as 3. The 10 jumps over the wall into the Tens box.
You CANNOT carry the full 10 across, because it then indicates 10 x Tens, which means 10 x 10, which equals 100. Therefore, only the 1 jumps across.
Hundreds
Tens 10
Units 3
1 × Tens means 1 × 10 = 10 If you want to indicate the place value of 13, it will look like this:
Hundreds
Tens 1
The same happens when there is a number greater than 9 in the Tens box.
Hundreds
Tens 84
23
Units 3
Units 2
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Since the 8 in the Tens box indicates 800, it must jump over the wall into the Hundreds box. It means 8 x 100 = 800.
Hundreds 8
Tens 4
Units 2
It is not always as easy or practical to draw boxes to determine place value and it can take a lot of time. There are three other methods to do the same without drawing boxes. Use the number 534.
Write down where each digit would go, without drawing the blocks. METHOD 1
5 Hundreds + 3 Tens + 4 Units METHOD 2 5 × 100 + 3 × 10 + 4 × 1 METHOD 3 500 + 30 + 4 These three methods are called expanded notation. When we write a number in expanded notation, we break down a number to show the place value of each digit. Expanded notation ‘shows’ place value.
ACTIVITY 3 1.
Write the number names.
Question
1.1
495
1.4
751
1.2 1.3 1.5 © Impaq
122 463 99
Answer Four hundred and ninety-five
One hundred and twenty-two Four hundred and sixty-three Seven hundred and fifty-one Ninety-nine
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Facilitator’s Guide G04 ~ Mathematics
2.
Unit
Write the numbers in the correct column in the place value table. Question
2.1
123
2.2
395
2.3
645
2.4
21
Answer
H
T
U
1
2
3
H
T
U
3
9
5
H
T
U
6
4
5
H
T
U
2
1
T
U
H 2.5 3.
1
4
4
Write the numbers in expanded notation. Use all three methods. Question
Answer
METHOD 1: 2H + 6T + 5U 3.1
265
METHOD 2: 2 × 100 + 6 × 10 + 5 × 1 METHOD 3: 200 + 60 + 5 METHOD 1: 1H + 2T + 3U
3.2
123
METHOD 2: 1 × 100 + 2 × 10 + 3 × 1 METHOD 3: 100 + 20 + 3 METHOD 1: 3H + 9T + 5U
3.3
395
METHOD 2: 3 × 100 + 9 × 10 + 5 × 1 METHOD 3: 300 + 90 + 5
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METHOD 1: 2T + 7U 3.4
27
METHOD 2: 2 × 10 + 7 × 1 METHOD 3: 20 +7 METHOD 1: 3U
3.5
4.
METHOD 2: 3×1 METHOD 3: 3
Give the value of the underlined numbers. Question
Answer
4.1
196
6
4.4
842
4.2 4.3 4.5
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3
542
500
154
50
90
92
800
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Facilitator’s Guide G04 ~ Mathematics
5.
Unit
1
Board game: Roll the dice and move the correct number of squares along on the board. Say the number of the block on which you land out loud. If you cannot say the number, move one block back. The first person to reach the finish block wins.
Start Finish 27
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Self-assessment Do the learners understand the work? Let them colour the faces that show what they can do. PLACE VALUE Requirements
Can the learners do it?
I can break down numbers into Hundreds, Tens and Units. I can give number names. I can indicate place value. I can do expanded notation (all three methods).
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Unit
1
Comparing whole numbers Use different sized physical objects to explain the concept of number comparison. You may also use place value – for example, 10 and 100 – to explain number comparison. Additional resources: •
100 charts
•
visit goo.gl/tVXMwP to find additional exercises and activities
• •
physical objects flash cards
When we compare numbers, we study the numbers to determine what is asked.
When you compare numbers, you must look at the value of the digits. Compare the following numbers: Circle the greatest number.
932 923 933
You already know how to compare numbers because you can order numbers. You also compare numbers when you complete sequences on number charts or number lines.
Example: Which whole numbers do the letters A and B represent?
25
A
29
31
B
Answer: A = 27 | B = 33 Example: Fill in the missing numbers on the number chart. Answer:
29
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You can now compare numbers, say which number is the greatest and which is the smallest,
and arrange them. But how do you indicate which number is the greatest or smallest without circling or underlining it?
We use symbols to indicate greater than, less than, and equal to. The symbols are:
<;>;=
To remember which way around the < and > signs go, just remember:
BIG > small small < BIG
The ‘small’ end always points to the smaller number. The ‘big’ end always points to the greater number.
<
3
932
OR
457
When two values are equal, we use the = to show that they are the same.
874 = 874
>
125
ACTIVITY 4 1.
Compare and arrange the numbers. The symbols (< and >) show whether the numbers must be arranged from small to great or from great to small. Question and answer
1.1
1.2
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89
218 453
192
118
>
181
<
654
295
29
>
118
<
295
30
385
>
218
<
192
89
453
>
181
<
385
29
654
Facilitator’s Guide G04 ~ Mathematics
1.3
2.
16
81
1.4
782
1.5
35
26
92
57
>
68
<
953
>
74
76
80
81
>
57
<
76
298
>
23
74
68
>
45
<
298
782
>
92
35
45
>
16
<
26
953
>
80
23
Provide the correct symbol (>, < or =) to indicate whether the following numbers are greater than, smaller than or equal to. 2.1
362 < 583
2.4
738 > 482
2.2 2.3 2.5 3.
1
Unit
Question and answer
183 < 362 200 > 178 679 < 697
Which whole numbers do the letters (A to G) represent? Question
A 3.1
13
3.2
310
B
14
17
18
19
Answer A = 15 B = 16
C 370
D
3.3
499
3.4
79
3.5
546
E
C = 340
D = 699 E = 899
999
F 82
F = 85
91
G 566
586
31
606
G = 556
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Facilitator’s Guide G04 ~ Mathematics
Self-assessment Do the learners understand the work? Let them colour the faces that show what they can do. COMPARING WHOLE NUMBERS Requirements
Can the learners do it?
I can arrange number sets from great to small. I can arrange number sets from small to great. I can complete missing numbers (in sequence and on number charts and number lines). I can use the correct symbols to indicate which numbers are greater than, smaller than or equal to other numbers (< , > or =).
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Facilitator’s Guide G04 ~ Mathematics
1
Unit
Representing whole numbers You learnt how to represent whole numbers in different ways. You learnt about: • number sets:
• visual representation of numbers:
{1; 2; 3; 4; ...}
• flow charts: • symbols:
• number lines: • place value: • expanded notation:
< ; > and =
H
T
U
METHOD 1: 9H + 0T + 2U METHOD 2: 9 × 100 + 2 × 1 METHOD 3: 900 + 2
Can you think of other representations?
ACTIVITY 5 1.
Choose any three representations to show the number 126. This is an open question and learners can represent or show the number 126 in any manner.
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LESSON 2: NUMBER SENTENCES Number sentences introduce operations with numbers (and in the senior and FET phases with variables). Once again, emphasise the concept using concrete, relevant examples. Learners in the intermediate phase sometimes work with number sentences in isolation. However, it is more common for learners to combine number sentences with other forms of representation; for example, mathematical problems that are simplified in words, as well as numbers and calculations that are represented in flow charts. It is important to include examples of mathematical problems that may be expressed as number sentences throughout the year. Also include number sentences in daily mental maths exercises. What are number sentences?
A number sentence is an arrangement of numbers and symbols: 3 + 9 = 12 55 – 6 = 49 8 × 5 = 40 12 ÷ 2 = 6
You will not always work with number sentences. Sometimes a maths problem is expressed in words, and before you can solve it mathematically, you must first formulate a number sentence for the problem.
Example: The neighbour’s dog had puppies. There are 4 females and 3 males. How many puppies are there in total?
Start by reading the problem again and writing down the information. Total number of puppies: 4 females plus 3 males.
The question is not asking you to count the females and males separately, therefore you must write the number sentence as follows: 4+3=?
Now it is easy to solve the problem:
4+3=7
This is an example of a number sentence in which you add numbers together. You can also write a number sentence with – , × , and ÷.
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Facilitator’s Guide G04 ~ Mathematics
1
Unit
Example: Refilwe loves to play with marbles. She has 12 marbles in her marble bag. During a competition, she loses 3 marbles. How many does she have left?
12 – 3 = ?
If you solve the number sentence:
12 – 3 = 9 Refilwe has 9 marbles left.
Write a few number sentences on your own.
ACTIVITY 6 1.
Write a number sentence for each of the statements or questions. Question
1.1 1.2 1.3 1.4 1.5
Mr Smith has 6 pencils. He picks up another 2 pencils on the playground. How many pencils does Mr Smith have altogether?
4 red birds sit on a branch. After a little while, 6 blue birds arrive and also sit on the branch. How many birds are sitting on the branch now? Lilly has 9 sweets. Lucy does not have any sweets. How many more sweets does Lilly have than Lucy?
Luke puts 15 flowers in a vase. 5 of the flowers wilt. How many flowers does Luke have left? Jo has 3 cards less than James. James has 18 cards. How many cards does Jo have?
Answer 6+2=8
4 + 6 = 10 9–0=9 15 – 5 = 10 18 – 3 = 15
Number sentences are not only represented with words, but can also be shown without using any words. You can solve and complete number sentences by: • inspection • trial and improvement • checking the solution by substitution
Let us look at solving number sentences by inspection.
The inspection method is another name for ‘guess the answer’. If you look at a number sentence and
must find the answer without making calculations, you will use the inspection method. In maths we only solve simple number sentences by inspection.
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Example: Study number sentence: 9 – 2 = We are not going to draw any pictures to solve this number sentence. Use mental maths and decide on an answer. You will then see – by using inspection – that the answer is 7.
ACTIVITY 7 1.
Solve the number sentences using inspection. 1.1 1.2 1.3 1.4 1.5
Question
Answer
273 + 6 =
279
42 – 12 =
30
67
62 + 5 =
964
997 – 33 =
42
54 – 12 =
Now let us look at the solving of number sentences through the trial and improvement method.
Using this method, you will guess an answer, but you will write it down and then calculate the number
sentence. If your answer is not correct, you can try another answer and calculate the number sentence until it is correct or until the number sentence is TRUE. This number sentence is FALSE:
This number sentence is TRUE:
413 – 2 = 410
413 – 2 = 411
Example: Study the number sentence below and use the trial and improvement method to solve it. 156 –
= 149
Attempt 1: Guess an answer that will make the number sentence TRUE. Guess that a 6 in the block will make the number sentence true.
156 – 6 = 149
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Unit
1
Attempt 2: To choose 6 as the answer will make the number sentence FALSE. Guess another answer. Since 6 is not enough to subtract from 156 to get 149, you must choose a greater number. Use the number 7.
156 – 7 = 149
Using this method, it may take a number of tries to solve the number sentence. It is time-consuming and not always the best method to use.
ACTIVITY 8 1.
Solve the number sentences using the trial and improvement method. 1.1 1.2 1.3 1.4 1.5
Question
Answer
458 – = 453
5
29 + = 143
114
15 + = 105
90
6
985 + = 991
110
326 – = 216
Checking the solution by substitution is very similar to the trial and improvement method. You have to study the number sentence and then decide on an answer. You then replace the answer in the number sentence and do the calculation. The number sentence must be TRUE. Number sentences are also a way of showing equivalence. What does equivalence mean? Equivalence is when one side of the number sentence is equal to the other side of the number sentence. In other words, showing that the number sentence is true.
The left side of the number sentence is equivalent (equal to) the right side of the number sentence. It means that the number sentence is true. Number sentences may also be used in flow charts and tables. Look at a few examples. 37
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Facilitator’s Guide G04 ~ Mathematics
In the flow chart below the number sentence is on the left-hand side and the equivalent value is on the right-hand side. Complete the flow chart so that all the number sentences have equivalent values.
260
4 + 256 156 – 7
19 + 199
149
=
218 445
86 + 359
909
999 – 90
Addition and subtraction in number sentences You already know that addition and subtraction are opposites. If you add 6 sweets to the 4 you already have, you will have 10 sweets. To return to what you originally had, you are going to do the opposite. Instead of adding 6 sweets, you are going to subtract them. The 10 sweets minus the 6 sweets gives you the original 4 sweets you had to start with. If you write this in a number sentence, it will look like this: Description in words
Number sentence
I have 4 sweets and add 6.
4 + 6 = 10
I have 10 sweets and subtract 6.
10 – 6 = 4
Can you think of more examples where addition and subtraction can be used as opposite calculations in number sentences? Let the learners write down more of their own examples in the space provided in the study guide. Learners must give their own examples. It is important that they go through the thought process – allow them to think of their own examples. Try to not help them, but you may guide them if they really need help; then let them think of more of their own examples.
You know that addition and subtraction are opposites and that they can also be used to
check the answers of number sentences. Study the examples below, which show the opposite effect of addition and subtraction: • 58 – 58 = 0 • 264 – 264 = 0 • 37 – 4 + 4 = 37 • 27 + 6 – 6 = 27
Write down another 5 of your own examples. © Impaq
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Unit
1
Learners must give their own examples. It is important that they go through the thought process – allow them to think of their own examples. Try to not help them, but you may guide them if they really need help; then let them think of more of their own examples.
1. 2.
Now that you have given your own examples, have another look at the number sentences shown above. What do you observe? Complete the sentences below.
When a number is subtracted from itself, the answer is 0.
When a number is added and then that same number is subtracted, you will get the number that you started with.
Pairs of equivalent number sentences The word ‘equivalent’ means that something is equal to something else or has the same value as something else.
Equivalent means equal to
Next you will ensure that two number sentences are equivalent and not merely a number sentence with an equivalent answer. Look at the examples below: Example 1 Number sentence 1: 36 + 13
You may use any of the methods in this unit to find the number sentence’s equivalent answer. Answer to number sentence 1: 49 Number sentence 2: 62 – 13
The equivalent answer of this number sentence is 49.
Do you see that both number sentences have the same answer? This means that the two number sentences are also the same, which means we can say that the two number sentences are equivalent. Number sentence 1 = Number sentence 2 36 + 13 = 62 – 13
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Facilitator’s Guide G04 ~ Mathematics
Example 2 Number sentence 1: 15 – 8
You may use any of the methods in this unit to find the number sentence’s equivalent answer. Answer of number sentence 1: 7 Number sentence 2: 15 + 8
The equivalent answer of this number sentence is 23.
Do you see that these number sentences DO NOT have the same answer? This means that the two number sentences are NOT the same; therefore we cannot say that the two number sentences are equivalent. The number sentences are not equal to ( ≠ ) each other:
Number sentence 1 ≠ Number sentence 2 15 – 8 ≠ 15 + 8
ACTIVITY 9 1.
Calculate the number sentences and determine if the number sentences are equivalent. Question
1.1
1.2
1.3
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Number sentence 1: 36 + 297 Number sentence 2: 297 + 36 Number sentence 1: 261 + 36 Number sentence 2: 297 – 36 Number sentence 1: 245 – 15 Number sentence 2: 249 – 19
Answer
36 + 297 = 333 297 + 36 = 333 Number sentence 1 = Number sentence 2 261 + 36 = 297 297 – 36 = 261 Number sentence 1 ≠ Number sentence 2 245 – 15 = 230 249 – 19 = 230 Number sentence 1 = Number sentence 2
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