Mathematics notes, unit 1
Numbers up to 10 000
Textbook Mathematics Year 3 (DLP), Part 1 and Part 2, pages Part 1, 1–26
What this chapter covers
- You can read and write numbers up to 10 000 in numerals and in words (1.1, 1.2)
- You can show a number with base ten blocks and an abacus, and say its place value and digit value (1.1, 1.4)
- You can partition a number by place value and by digit value (1.4)
- You can compare numbers and arrange them in ascending and descending order (1.1, 1.3)
- You can count on and count back, and complete number patterns (1.3, 1.7)
- You can estimate using "more than", "less than" and "more or less" (1.5)
- You can round off numbers to the nearest ten, hundred and thousand (1.6)
- You can solve problems about numbers up to 10 000 (1.8)
1.1Reading and writing numbers
In Year 3 you learn numbers all the way up to 10 000 (ten thousand). You see big numbers everywhere: a chocolate factory makes 2 780 packets a day, and Kuala Lumpur Bird Park has about 3 000 birds.
A four-digit number has four digits, for example 2 780. We leave a small space after the thousands digit so the number is easy to read.
To read a number, start from the left. Say the thousands, then the hundreds, then the tens and ones.
| Numeral | In words |
|---|---|
| 2 780 | two thousand seven hundred and eighty |
| 1 130 | one thousand one hundred and thirty |
| 3 589 | three thousand five hundred and eighty-nine |
| 9 027 | nine thousand and twenty-seven |
| 3 006 | three thousand and six |
| 10 000 | ten thousand |
A 0 inside a number is not said out loud. 3 006 is "three thousand and six", not "three thousand zero hundred zero six". But when you write the numeral, you must write the 0s: four thousand and sixty is 4 060, not 460 or 4 600.
Counting in thousands
10 hundreds make 1 thousand. When we count in thousands we go 1 000, 2 000, 3 000, 4 000, 5 000 … all the way to 10 000. Ten thousands make ten thousand.
Did you know? 10 000 is the first number with five digits. It is 1 more than 9 999, the largest four-digit number.
Write "eight thousand two hundred and three" in numerals. Then write 1 642 in words.
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8 203. 1 642 is one thousand six hundred and forty-two.
1.2Place value, digit value and partitioning
Every digit sits in a place. The place tells us how much the digit is worth.
Place value is the name of the place: thousands, hundreds, tens or ones.
Digit value is how much the digit is worth in that place. The 2 in 1 247 is in the hundreds place, so its digit value is 200.
Digit value is how much the digit is worth in that place. The 2 in 1 247 is in the hundreds place, so its digit value is 200.
| Digit | 1 | 2 | 4 | 7 |
|---|---|---|---|---|
| Place value | thousands | hundreds | tens | ones |
| Digit value | 1 000 | 200 | 40 | 7 |
Partitioning
To partition a number means to break it into parts. We can do it in two ways.
Worked example: partition 1 247 and 6 035
Step 1. By place value: 1 247 = 1 thousand + 2 hundreds + 4 tens + 7 ones.Step 2. By digit value: 1 247 = 1 000 + 200 + 40 + 7.
Step 3. 6 035 has a 0 in the hundreds place. By place value: 6 thousands + 0 hundreds + 3 tens + 5 ones.
Step 4. By digit value we can leave out the 0: 6 035 = 6 000 + 30 + 5.
Reading an abacus
On the abacus, the beads that touch the middle bar are counted. An upper bead is worth 5. Each lower bead is worth 1.
So 7 812 = 7 000 + 800 + 10 + 2.
The parts can be written in any order, but each digit value must match its place. "7 000 + 10 + 800 + 2" is still 7 812. "7 + 8 + 1 + 2" is wrong: those are digits, not digit values.
In 2 917, what is the place value of 9? What is the digit value of 2?
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9 is in the hundreds place. The digit value of 2 is 2 000.
1.3Comparing and arranging numbers
At a Hari Kebangsaan sale, a stall sold 967 caps and 1 835 flags. Which was sold more?
How to compare two numbers
Step 1. Count the digits. A four-digit number is larger than a three-digit number. 1 835 > 967, so more flags were sold.Step 2. Same number of digits? Compare the thousands first. 1 835 and 1 249 both have 1 thousand.
Step 3. Equal? Compare the hundreds. 2 hundreds is smaller than 8 hundreds, so 1 249 is smaller than 1 835.
Step 4. Still equal? Move on to the tens, then the ones.
A number line helps too. The more to the right, the larger the number.
Ascending and descending order
Ascending order: from the smallest to the largest (going up a staircase).
Descending order: from the largest to the smallest (going down).
Descending order: from the largest to the smallest (going down).
Worked example: trees in a replanting project
The numbers are 3 000, 3 100, 2 800 and 2 900.Step 1. Find the smallest: 2 800. Find the largest: 3 100.
Step 2. Ascending order: 2 800, 2 900, 3 000, 3 100.
Step 3. Descending order: 3 100, 3 000, 2 900, 2 800.
Do not compare only the last digit. 4 509 is smaller than 4 590, even though 9 is bigger than 0. Look at the tens first: 0 tens is less than 9 tens.
Which number is smaller: 3 408 or 3 296?
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The thousands are equal (3). Hundreds: 2 is less than 4. So 3 296 is smaller.
1.4Counting and number patterns
We can count on (the numbers get bigger) or count back (the numbers get smaller) in ones, twos, fives, tens, hundreds, thousands and more.
- Count on in ones from 5 821: 5 821, 5 822, 5 823, 5 824
- Count on in tens from 5 821: 5 821, 5 831, 5 841, 5 851
- Count back in fives from 5 840: 5 840, 5 835, 5 830, 5 825
- Count on in hundreds: 5 730, 5 830, 5 930, 6 030, 6 130
- Count back in thousands: 7 930, 6 930, 5 930, 4 930, 3 930
A number pattern is a list of numbers that follows a rule. Say the rule like this: "The number pattern is increasing in fives" or "decreasing in sixes".
Worked example: find the missing numbers
3 076, 3 070, 3 064, ___, 3 052Step 1. Find the rule: 3 076 − 3 070 = 6. The numbers go down by 6 each time.
Step 2. 3 064 − 6 = 3 058.
Step 3. Check: 3 058 − 6 = 3 052. Correct! The pattern is decreasing in sixes.
Did you know? A bank book shows a pattern too. If Pak Ali saves RM1 000 each month, his balance goes RM4 070, RM5 070, RM6 070, RM7 070, RM8 070. The pattern is increasing in thousands.
Complete: 6 269, 6 260, ___, 6 242. State the pattern.
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6 260 − 9 = 6 251. The pattern is decreasing in nines.
1.5Estimating
To estimate means to make a good guess without counting or measuring everything. We use the words more than, less than and more or less (about).
Worked example
A full bottle of shampoo has 1 200 ml. Another bottle of the same size is about half full.Step 1. Half of 1 200 ml is 600 ml.
Step 2. So the second bottle has more or less 600 ml.
Use something you know as a guide. If one ribbon is 100 cm and a red ribbon is about half as long, the red ribbon is more or less 50 cm.
1.6Rounding off
Sometimes we do not need the exact number. We round off to a "nice" number that is near it. A news report may say "about 6 000 cm" instead of "5 820 cm".
To the nearest ten
So 1 362 rounded off to the nearest ten is 1 360.
To the nearest hundred
A number in the middle is rounded up to the larger number. So 1 350 to the nearest hundred becomes 1 400.
To the nearest thousand
Worked example: the Rawang Bypass bridge is 5 820 cm high
Step 1. Look at the hundreds digit: 5 820. It is 8.Step 2. The rule: if the hundreds digit is 5, 6, 7, 8 or 9, add 1 to the thousands. If it is 0, 1, 2, 3 or 4, keep the thousands the same.
Step 3. 8 is in the first group, so 5 thousands becomes 6 thousands.
Step 4. Change the hundreds, tens and ones to 0: 6 000.
Mount Kinabalu is 4 095 metres high. The hundreds digit is 0, so we keep 4 thousands: 4 095 rounds off to 4 000.
| Round off to the nearest … | Look at this digit | Example: 8 519 |
|---|---|---|
| ten | ones | 8 520 |
| hundred | tens | 8 500 |
| thousand | hundreds | 9 000 |
Round off in one step only. 3 482 to the nearest thousand is 3 000 (hundreds digit 4). Do not round to 3 500 first and then to 4 000.
Round off 7 302 to the nearest ten, hundred and thousand.
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Nearest ten: 7 300 (ones digit 2). Nearest hundred: 7 300 (tens digit 0). Nearest thousand: 7 000 (hundreds digit 3).
1.7Solving problems
Read the problem carefully. Find the important information. Then choose a way to solve it: make a table, look for a pattern, draw a number line or work backwards.
Make a table: Juli scored 2 605 and Stacy scored 2 719
| thousands | hundreds | tens | ones | |
|---|---|---|---|---|
| Juli | 2 | 6 | 0 | 5 |
| Stacy | 2 | 7 | 1 | 9 |
Look for a pattern: chalet numbers
The chalets are numbered 1 302, 1 304, 1 306 … (increasing in twos). The purple chalet is the 6th one.1 302, 1 304, 1 306, 1 308, 1 310, 1 312. The purple chalet is number 1 312.
Work backwards: about 1 500 people
"Rounded off to the nearest ten, the audience was 1 500." What could the real number be?Step 1. Draw a number line from 1 490 to 1 510. Numbers from 1 495 up to 1 504 are nearer to 1 500 (1 495 is in the middle, so it is rounded up).
Step 2. The number became 1 500 after rounding, so it was not 1 500 to start with. The textbook answer: the audience could be 1 495 to 1 499 or 1 501 to 1 504.
Amir uses the cards 0, 4, 5 and 8 to make a four-digit number with 5 in the hundreds place. Give one possible number.
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Any one of 4 508, 4 580, 8 504, 8 540. (The 0 cannot go first, or it is not a four-digit number.)
Summary
- Numbers up to 10 000 are read from the left: thousands, hundreds, tens, ones. A 0 is not said, but it must be written.
- Place value is the name of the place; digit value is what the digit is worth (the 2 in 1 247 is worth 200).
- Partition by place value (1 thousand + 2 hundreds + 4 tens + 7 ones) or by digit value (1 000 + 200 + 40 + 7).
- To compare, look at the thousands first, then hundreds, tens and ones. On a number line, larger numbers are on the right.
- Ascending = smallest to largest. Descending = largest to smallest.
- A number pattern is "increasing in …" or "decreasing in …". Find the rule by subtracting two numbers next to each other.
- Estimate with "more than", "less than" and "more or less".
- Rounding off: look at the digit to the right of the place you want. 5 or more, round up; 4 or less, keep it.
- Solve problems by making a table, looking for a pattern, drawing a number line or working backwards.