advanced edition sample - understandingmaths.com...and of course the graded exercises at the ......

iUnderstanding Year 3 Maths Advanced Edition

Warwick Marlin © Five Senses Education

CLEAR AND CONCISE

GRADED EXERCISES

SOLUTIONS AT BACK

ALL ABILITY GROUPS

VERY USER FRIENDLY

�

�

�

�

�

ADVANCED ED IT ION

UNDERSTANDING

Y E A R 3

Warwick Marlin B.Sc. Dip.Ed.

M A T H

Author

S

SAMPLE

vvUnderstanding Year 3 Maths Advanced Edition


Page

CONTENTS

INTRODUCTION vi

NUMBER AND ALGEBRA

NUMBER AND PLACE VALUE (PART 1) 1

NUMBER AND PLACE VALUE (PART 2) 27

FRACTIONS AND DECIMALS 55

MONEY AND FINANCIAL MATHEMATICS 77

PATTERNS AND ALGEBRA 93

MEASUREMENT & GEOMETRY

USING UNITS OF MEASUREMENT 109

SHAPE 133

LOCATION AND TRANSFORMATION 149

GEOMETRIC REASONING 163

STATISTICS & PROBABILITY

STATISTICS 176

PROBABILITY 183

SOLUTIONS TO GRADED EXERCISES 197

NOTE: The New Australian National Curriculum has been split into 3 major strands:

A Number & Algebra B Measurement & Geometry C Statistics & Probability

In the Year 3 content descriptions, these 3 major strands have been further subdivided into the 10 chapters shown above.

�

�

�

�

�

�

�

�

�

�

�

SAMPLE

ixUnderstanding Year 3 Maths Advanced Edition


HOW CAN THIS BOOK HELP?

1) Firstly, the book has been split up into the 10 MAJOR IMPORTANT TOPICS.

2) Each of these major topics has been broken down into a number of simpler ideas and rules, thus saving educators and students valuable TIME in research.

3) Each page covers only one idea or rule, thus giving students CLARITY.

4) Each concept is thoroughly, but simply, explained for UNDERSTANDING.

5) Each formula, rule or set of steps is highlighted in larger print for ease of MEMORISING.

6) Each formula and page contains at least one fully worked example. This is not only reinforces understanding, but also shows the student how to APPLY each formula to typical questions.

7) At the end of each chapter, there are usually 6 or more comprehensive graded exercises for PRACTICE, which cover all the ideas in the topic. If a student is not sure how to do a particular problem, all they need to do is turn back to the page number shown in order to find a very similar example. (Please, read more about these exercises in the next section.)

8) In addition, the text presentation is well spaced out, in order to make the subject matter more APPEALING and USER FRIENDLY to this age group.

As you can see this book should prove to be an invaluable teaching aid for teachers, coaches and tutorial centers, because it thoroughly summarises the 10 major topics which are the basis of the Australian Curriculum.

It will be very beneficial to parents, because it will provide them with a very structured and clear idea of the core syllabus, and what their children should know by the end of Grade 3. If their child has a particular problem (say on adding fractions) it is very easy to find the page and explanations relating to that idea – and hence help their child.

Most important of all, it will prove to be an excellent reference for students of all abilitygroups. The user friendly format and layout makes it very much faster for a pupil to thoroughlymaster one major topic in a relatively short period of time, because it is so easy to see how each idea is linked to the previous one. It has all the rules and corresponding examples clearlyset out topic by topic, page by page. In addition it teaches the pupils to read explanations, as well as to look back and research similar problems. And of course the graded exercises at the end of each topic chapter will help students of all abilities to practise and apply their knowledge.It is so easy for students to work through the book by themselves with the minimum of supervision and help.

1. Brief concise explanation.

2. Rule or formula or set of steps in larger print.

3. Fully worked examples.

Each page explains only ONE simple important

idea or formula as shown on the left.

SAMPLE

xUnderstanding Year 3 Maths Advanced EditionWarwick Marlin © Five Senses Education

HOW TO USE THIS BOOK EFFECTIVELY?

As stated in the previous section, each chapter has been carefully split up into all the important ideas relating to that particular major topic. After carefully reading through and understanding a chapter, the next stage is for the student to work through some of the revision exercises given at the end. These have been carefully graded into different levels of difficulty, and most Grade 3 students should try to complete the first 3 levels. Remember that completing a level involves careful marking of your answers from the solutions given at the back of the book. If you got any questions wrong, then it is very important to find out why you got them wrong, before you move onto the next level.

Note: Most chapters have six graded exercises at the end, but longer chapters may have 7 or 8 graded exercises. The reference pages work in the following way. For example: 117, 178 in the margin means turn back to page 177 and page 178. However 173-176 means turn back to pages 173, 174, 175 and 176.

EASIER QUESTIONS

These questions given in level 1 (sometimes also in level 2) are intended to build up confidence. Most of the questions are almost identical to the examples given throughout the chapter – except with the numbers changed. If the student does have difficulty, then he/she can easily refer back to a similar example on the page number shown in the margin.

AVERAGE QUESTIONS

The questions in Level 2 and Level 3 are of average difficulty level, and will give all students (weak, average & gifted) a good opportunity to practice and consolidate the ideas and rules given throughout most of the chapter. Once again, if a student does have difficulty with a particular problem, then he/she can quickly refer back to a similar example on the page number shown. All students should try to complete and understand the questions in the first 3 levels.

HARDER QUESTIONS

The questions in this level are more difficult than the ones in level 2 and 3. The numbers involved are larger, and some of the more difficult ideas in the topic are tested. Reference page numbers have NOT been included in the margin, in order to make students who reach this level more skillful in researching through the chapter for their own information. Level 4 should be done by more capable students, or those students who have found earlier levels straightforward.

PROBLEM SOLVING

This more difficult level has been included to challenge those students who are more gifted at Maths. Usually the questions are more sentence and problem oriented, and therefore they involve more reading and comprehension skills. It is unlikely that any of the questions in this level can be done mentally, because several different ideas, rules or steps are usually required.

All students should try to complete at least the first

3 levels. The last 2 levels are aimed at keener and more

gifted students.

SAMPLE

2Understanding Year 3 Maths Advanced EditionWarwick Marlin © Five Senses Education

INTRODUCTION

In some schools wooden or plastic blocks are used to give students a visual idea of the Base-10 number system that we use.

A single cube represents one unit which is called a SHORT.

Ten shorts joined together represents 10 units which is called a LONG.

Ten longs joined together represents 100 units which is called a FLAT.

Ten flats joined together represents 1 000 units which is called a BLOCK.

Example: What number do the following Base-10 materials represent?

Solution: 3 blocks + 2 flats + 5 longs + 8 shorts = (1 000 + 1 000 + 1 000) + (100 + 100) + (10 + 10 + 10 + 10 + 10) + (1 + 1 + 1 + 1 + 1 + 1 + 1+ 1) = 3 000 + 200 + 50 + 8 = 3 258

OR

Note: We can easily recognise the blocks, flats, longs, and shorts without having to write all the detailed lines.

The above materials are called Base-10 materials and are very useful in helping students to understand the HINDU-ARABIC number system that we use today. A long is 10 times larger and heavier than a short, and a flat is 10 times larger and heavier than a long, and a block is 10 times larger and heavier than a flat.

+ + +

Visual (seeing) and tactile (touching)

experience is important for understanding.

+ + +

SAMPLE

3Understanding Year 3 Maths Advanced Edition


THE HINDU-ARABIC NUMBER SYSTEM

The number system that we use today is called the Hindu-Arabic number system because it was first developed in India and Arabia over 1 000 years ago. This system uses the 10 digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and it is based on multiples of 10 with zero as a ‘place holder’.

Example: Find the place value of 8 in each of the following:

a) 3 718 b) 3 781 c) 3 871 d) 8 371

Solutions: a) The place value of 8 is 8 × 1 = 8 b) The place value of 8 is 8 × 10 = 80 c) The place value of 8 is 8 × 100 = 800 d) The place value of 8 is 8 × 1 000 = 8 000

Important note:

Example: Write fifty-seven thousand, six hundred and eighty-two

a) using a comma b) using a space

Solutions: a) 57,682 b) 57 682

The Hindu-Arabic number system uses a place value system.Each place value is 10 times greater than the place value directly to its right.The value of any digit in a number depends on which position (or place) it sits in that number.

The value of 8 depends on its place in the

number.

Some teachers and some Maths text books will place a ‘comma’ to separate every 3 digits in a larger whole number.Other teachers and texts will use a ‘space’ to separate every 3 digits in a larger whole number.Both methods are correct and you should use the method explained to you by your teacher.

In this book we will be using a ‘SPACE’ as a

separator.

SAMPLE


PLACE VALUE

Since this page deals with WHOLE NUMBERS, the first importantidea is to understand the value of each of the different columns.As we explained on the previous page, our number system todayis based on the Hindu-Arabic system where the VALUE of a number is determined by its PLACE in a particular column.

Example: Look at the place value chart (or table) below.

The place value of 8 is 8 × 1 = 8 The place value of 3 is 3 × 10 = 30 The place value of 0 is 0 × 100 = 0 The place value of 6 is 6 × 1 000 = 6 000 The place value of 5 is 5 × 10 000 = 50 000

In Grade 3 you will learn 3 important ways of describing a whole number:1. AS AN ORDINARY NUMERAL: 56 038

2. IN WORDS: Fifty-six thousand and thirty-eight

3 IN EXPANDED NOTATION: (5 × 10 000) + (6 × 1 000) + (0 × 100) + (3 × 10) + (8 × 1)

Example: a) b) c)

For each abacus above, write the numeral and also write the numeral in words.

Solutions: a) Numeral = 3 524 In words: Three thousand, five hundred and twenty-four

b) Numeral = 5 143 In words: Five thousand, one hundred and forty-three

c) Numeral = 4 072 In words: Four thousand and seventy-two

Th H T U Th H T U Th H T U

In Grade 3, students are required to read and write numbers upto 'ten

thousands'.

As you can see, there are NO hundreds in this number. However the zero must still be written down as a 'place'

holder.

Ten Thousands Thousands Hundreds Tens Ones or Units10 000 1 000 100 10 1

5 6 0 3 8

You must be able to change from one notation to another.

SAMPLE

5Understanding Year 3 Maths Advanced Edition


READING AND WRITING WHOLE NUMBERS

There are several different ways of reading and writing numbers which is best understood by looking at the examples below.

Example 1: Write the numeral 67 832 using words.

Solution: Sixty-seven thousand, eight hundred and thirty-two

Example 2: Write the number 2 938 using 'standard' and 'non standard' words.

Solution: Two thousand, nine hundred and thirty-eight Standard form Twenty-nine hundred and thirty-eight Non standard form

Note: Two thousand, nine hundred = Twenty-nine hundred

Example 3: Calendar years are also usually read and written differently.

1968 in words ‘nineteen sixty-eight’ NOT one thousand, nine hundred and sixty-eight

However, 2017 in words ‘two thousand and seventeen’ OR ‘twenty seventeen’ are both correct. Note: When writing yearly dates, we do not place spaces between the digits.

Example 4: When we write seven hundred and six in standard notation, why is the zero between the digits 7 and 6?

Solution: In 706 the zero acts as a ‘place holder’. If we did not place a zero in the 10's column then the number would read 76 or seventy-six. And 76 is not the same as 706!

When writing a number in words a comma is placed after the word ‘thousand’. The word ‘and’ is written after the word ‘hundred’ - if there is no ‘hundred’ word, then ‘and’ is written after the word ‘thousand’. Also a hyphen (-) is used for numbers from 21 to 99.

Numbers with 4 or more digits may be read in a NON STANDARD way. We can write the number in hundreds without mentioning the word 'thousand'.

SAMPLE

Note: Only turn back to page number shown if you have difficulty. Page


EASIER QUESTIONSLEVEL 1 — NUMBER AND PLACE VALUE (Part 1)

Q1. Count the Base-10 blocks below and write the number as a numeral and also in words. 2 a) b)

Q2. Write the following as ordinary numerals. 4 a) Five hundred and seventy-nine = b) Two thousand, four hundred and thirty-eight =

Q3. Order the numbers from smallest to largest. 9 a) 1 035, 627, 2 448, 962, 456 ____________________________ b) 3 258, 1 792, 599, 2 871, 836 ____________________________

Q4. Write the largest possible numbers using the digits given. 4 a) 2, 3, 1 b) 3, 4, 9

Q5. How many digits are there in each number? Write your answer on the line. 9 a) 13 b) 126 c) 649 d) 1 587

Q6. Write the place value of the underlined digits. 4 a) 87 b) 3 69 c) 856 d) 1 246 e) 2 943 f) 4 513

Q7. Write the following in numerals. 4, 6 a) (7 × 10) + (8 × 1) = b) (2 × 100) + (7 × 10) + (5 × 1) = c) (1 × 1 000) + (3 × 100) + (5 × 10) + (4 × 1) = d) (3 × 1 000) + (6 × 100) + (2 × 10) + (8 × 1) =

Q8. Write the numerals below in expanded notation. 4, 6 a) 84 = b) 432 = c) 1 246 = d) 2 461 =

Q9. Is 36 closer to 30 or 40? 36 is closer to 40, therefore it is rounded up to 40. Try this! 7 Round off the following numbers to the nearest 10. a) 48 _____ b) 53 _____ d) 85 _____ e) 121 _____

Q10. First, round off the numbers to the nearest 10, and then estimate the answers. 7, 19 a) 34 + 17 b) 46 + 62 c) 128 + 91 + Round off + Round off + Round off

Estimate Estimate Estimate

Remember!The number 5 is

always rounded up.SAMPLE

Note: Only turn back to page number shown if you have difficulty. Page


AVERAGE QUESTIONS

Q1. Look at the following letter cards: 11 If the first letter is A, what is the

a) third letter? b) sixth letter? c) second last letter? d) the fourth letter?

Q2. Write the following numerals in expanded notation: 4, 6

a) 928 b) 367 c) 4 316 d) 37 049

Q3. Fill in the boxes with the next even numbers. 10

a) 12, b) 50, c) 234,

d) 890, e) 1 256, f) 3 578,

Q4. Fill in the boxes with the next odd numbers. 10

a) 21, b) 99, c) 561,

d) 745, e) 2 707 f) 4 873,

Q5. Use > OR < to compare the numbers. 8

a) 48 39 b) 81 97

c) 372 427 d) 785 628

Q6. Write the smallest possible numbers using the given digits. 9 a) 5, 1, 3 b) 4, 2, 6

Q7. Order the numbers from largest to smallest. 9 a) 3 847, 2 916, 5 467, 1 929, 6 310 b) 2 850, 5 619, 8 374, 4 956, 3 798

Q8. Write the value of the underlined digits. 4 a) 720 b) 1 384 c) 5 566 d) 2 423 e) 9 475 f) 7 482

Q9. Write the numbers below as numerals. 4, 6 a) (9 × 100) + (7 × 10) + (3 × 1) = b) (2 × 1 000) + (8 × 100) + (5 × 10) + (4 × 1) = c) (4 × 1 000) + (2 × 100) + (1 × 10) + (8 × 1) = d) (6 × 1 000) + (1 × 100) + (7 × 10) + (5 × 1) =

20 > 15 Twenty is greater

than fifteen.38 < 52

Thirty-eight is less than fifty two.

A EC GB FD

LEVEL 3 — NUMBER AND PLACE VALUE (Part 1)

SAMPLE


HARDER QUESTIONS

Q1. Write the following numbers in words: a) 739 b) 4 631 c) 23 486 d) 32 045

Q2. What is the value of 7 in the following numbers: a) 7 349 b) 15 702 c) 23 475 d) 71 639

Q3. Order the following numbers from highest to lowest: a) 6 345, 6 453, 6 534, 6 354, 6 435, 6 543 b) 7 219, 7 291, 7 912, 7 192, 7 921, 7 129

Q4. Find the sum: a) 14 326 b) 13 413 c) 15 347 d) 32 783 2 531 4 251 1 536 9 419 + 102 + 2 132 + 452 + 6 342

Q5. Find the difference.

a) 3 475 b) 25 475 c) 39 325 d) 56 321 – 683 – 6 923 – 6 147 – 27 453

Q6. Fill in the circles with the correct number and then give the sum. a) 5 + (2 + 4) = (5 + 2) + = b) 8 + 11 = 11 + =

c) (6 + 7) + 2 = + (7 + 2) = d) + (4 + 2) = (3 + 4) + 2 =

e) 0 + 135 = 135 + = f) 27 + 13 = + 27 =

Q7. Change the order and then find the sum without using pen and paper.

a) 130 + (70 + 24) = b) (65 + 13) + 87 =

Q8. Each letter stands for a numeral. Find the value of each letter.

a) AAB b) NMN c) PQRS + BAB + NNM + PPQR 648 4 5 5 2 3 5 7

Q9. I am an odd number with four digits. I have 5 in the hundreds place and 1 in the tens place. I am a number greater than 2 500 but less than 3 000. The sum of my ones place number and tens place number is 10. What number am I? ______

Q10. I am a 4-digit number. The digit in my ones place is the largest even number and the digit in my hundreds place is the smallest odd number. The digit in my thousands place is 1 less than the digit in my ones place. The sum of all my digits is 21. What number am I? ______


SAMPLE


Q1. Maria collected 1 560 stamps. Lourdes and Francisca each collected 3 842 stamps. How many stamps have Maria, Lourdes and Francisca collected altogether?

Q2. Jolie’s bakeshop makes 5 430 cupcakes everyday. 1 850 cupcakes are sent to Jamaica Coffee shop and 1 900 cupcakes are sent to Zhi’s Coffee House. How many cupcakes are left in the bakeshop?

Q3. Khan and his friends went strawberry picking. Khan picked 2 800 strawberries. His friend Allan picked half the number Khan picked. John picked 485 more strawberries than Allan. How many strawberries have the three friends picked altogether?

Q4. Chang had 2 345 marbles. He put 892 marbles in a small box and put the remaining marbles in a bigger box. He accidentally dropped the bigger box and lost 376 marbles. How many marbles were left in the bigger box?

Q5. Truck A carries 2 350 watermelons. Truck B carries 2 000 more watermelons than truck A, and truck C carries 1 170 less watermelons than truck B. a) Which truck carries the most watermelons? b) Which truck carries the least watermelons? c) Round off the number of watermelons to the nearest 100, and then estimate the total number of watermelons from the 3 different trucks. What is the estimated total number of watermelons from the three different trucks?

Q6. My favourite number is a 4-digit number. The digit in my thousands place is the largest odd number and the digit in my ones place is the smallest odd number. The digits in the tens place and hundreds place are the same. The total of all digits when added together is 16. What’s my favourite number?

Q7. Which is the third highest odd number that can be formed from the 4 digits 3, 7, 9 and 4?

Q8. Richard is building a small wall at the front of his house. He bought 4 500 bricks, and he used 3 876 of them. How many bricks are left? Q9. My sister is twice my age. If our ages add upto 42, what is my age?

Q10. I am a 5 digit number. The number in the thousands column is the highest even number. The number in the tens column is the second lowest odd number. The number in the ten thousands column is three times the number in the tens column. The number in the units column is 6, and the number in the hundreds column is one third of the number in the units column. What number am I?

Q11. I am a 4 digit number less than 2 000. If all my digits are added together, the answer is 19. The sum of my thousands place number and tens place number is 10. The digit in my hundreds place is twice the digit in my ones place. What number am I?

PROBLEM SOLVING

8701

234 56 9


SAMPLE


LEVEL 1 – Number and Place Value (Addition and Subtraction)

Q1. a) 823 b) 1 149 Eight hundred and twenty-three One thousand, one hundred and forty-nine Q2. a) 579 b) 2 438 Q3. a) 456, 627, 962, 1 035, 2 448 b) 599, 836, 1 792, 2 871, 3 258 Q4. a) 321 b) 943 Q5. How many digits are there in each number? Write your answer on the line. a) 13 2 b) 126 3 c) 649 3 d) 1 587 4 Q6. Write the place value of the underlined digits. a) 87 7 b) 3 69 300 c) 856 50 d) 1 246 1 000 e) 2 943 3 f) 4 513 4 000 Q7. Write the following in numerals. a) (7 × 10) + (8 × 1) = 78 b) (2 × 100) + (7 × 10) + (5 × 1) = 275 c) (1 × 1 000) + (3 × 100) + (5 × 10) + (4 × 1) = 1 354 d) (3 × 1 000) + (6 × 100) + (2 × 10) + (8 × 1) = 3 628 Q8. Write the numerals below in expanded notation. a) 84 = (8 × 10) + (4 × 1) b) 432 = (4 × 100) + (3 × 10) + (2 × 1) c) 1 246 = (1 × 1 000) + (2 × 100) + d) 2 461 = (2 × 1 000) + (4 × 100) + (4 × 10) + (6 × 1) (6 × 10) + (1 × 1) Q9. Is 36 closer to 30 or 40? 36 is closer to 40, therefore it is rounded up to 40. Try this! Round off the following numbers to the nearest 10. a) 48 50 b) 53 50 c) 74 70 d) 85 90 e) 121 120 f) 639 640 Q10. First, round off the numbers to the nearest 10, and then estimate the answers. a) 34 + 17 b) 46 + 62 c) 128 + 91

30 + 20 Round off 50 + 60 Round off 130 + 90 Round off

50 Estimate 110 Estimate 220 Estimate

LEVEL 2 – Number and Place Value (Addition and Subtraction)

Q1. a) 483 Four hundred and eighty-three b)935Ninehundredandthirty-five c) 6 217 Six thousand, two hundred and seventeen Q2. Shade the even numbers and cross out the odd numbers.

Q3. Shade the addition and subtraction facts that are equal to 50.

Q4. a) 45 + 57 = 102 b) 128 – 26 = 102 c) 215 + 43 = 258 d) 319 – 205 = 114

Q5. a) b) c) d)

Remember!The middle number 5 is always rounded up.

Remember! Numbers that can be divided by 2 are even

numbers and numbers than can’t be divided by

2 are odd numbers.

33

318

28

77

90

155

57

64

142

561

28 + 38 83 – 38 48 + 1220 + 30 38 + 12 63 – 13

78 + 18 72 – 32 85 – 3558 – 8 15 + 45 28 + 38

Th H T O 3 4 7 2+ 6 3 1 5 9 7 8 7

Th H T O 2 5 8 1+ 3 0 1 6 5 5 9 7

Th H T O 3 5 4 9+ 3 3 0 3 8 7 9

Th H T O 3 7 3 4+ 2 1 6 3 5 8 9 7

SAMPLE

advanced edition sample - understandingmaths.com...and of course the graded exercises at the ......

Documents