Mathematics notes, unit 1
Numbers and Operations
Textbook Mathematics Year 4 (DLP), pages 1–66
What this chapter covers
- You can read and write numbers up to 100 000 in numerals and in words (1.1)
- You can state the place value and digit value of a digit, and partition numbers (1.1)
- You can compare numbers and arrange them in ascending and descending order (1.1)
- You can tell even numbers from odd numbers (1.2)
- You can estimate quantities and round off numbers to the nearest ten, hundred, thousand and ten thousand (1.3, 1.4)
- You can complete and state number patterns (1.5)
- You can add and subtract numbers up to 100 000 and check by estimating (1.6)
- You can multiply by 10, 100 and 1 000 and by two-digit numbers, and divide by 10, 100, 1 000 and two-digit numbers (1.6)
- You can solve mixed operations, find unknown values and solve problems (1.7, 1.8, 1.9)
1.1Reading, writing and comparing numbers
In Year 4 you work with numbers up to 100 000. A five-digit number is read in two groups: the thousands group first, then the last three digits. A football match can have 62 307 people watching it.
A five-digit number has ten thousands, thousands, hundreds, tens and ones. We leave a small space after the thousands digit: 62 307.
| Numeral | In words |
|---|---|
| 13 701 | thirteen thousand seven hundred and one |
| 47 293 | forty-seven thousand two hundred and ninety-three |
| 20 008 | twenty thousand and eight |
| 50 813 | fifty thousand eight hundred and thirteen |
A 0 is not said, but you must write it. "Twenty thousand and four" is 20 004, not 2 004 or 20 400.
Place value, digit value and partitioning
| Digit | 6 | 2 | 3 | 0 | 7 |
|---|---|---|---|---|---|
| Place value | ten thousands | thousands | hundreds | tens | ones |
| Digit value | 60 000 | 2 000 | 300 | 0 | 7 |
Place value is the position of a digit in a number. Digit value is what the digit is worth in that place. In 62 307, the 6 is in the ten thousands place and its digit value is 60 000.
Worked example: partition 62 307
By place value: 62 307 = 6 ten thousands + 2 thousands + 3 hundreds + 0 tens + 7 ones.By digit value: 62 307 = 60 000 + 2 000 + 300 + 7. A 0 can be left out.
What is the digit value of 8 in 48 106? Write 48 106 in words.
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8 is in the thousands place, so its digit value is 8 000. 48 106 is forty-eight thousand one hundred and six.
Comparing and arranging
Compare from the left: ten thousands first, then thousands, hundreds, tens and ones. Stop at the first digit that is different.
Worked example
Numbers: 15 367, 13 719, 15 202, 14 419.Ascending order (smallest to largest): 13 719, 14 419, 15 202, 15 367.
Descending order (largest to smallest): 15 367, 15 202, 14 419, 13 719.
Do not look only at the last digits. 65 090 is smaller than 65 501 even though 90 looks big: compare the hundreds first (0 against 5).
1.2Even and odd numbers
An even number ends in 0, 2, 4, 6 or 8 and can be divided by 2 with no remainder. An odd number ends in 1, 3, 5, 7 or 9 and has a remainder of 1 when divided by 2.
Worked example: 32 and 85
32 ÷ 2 = 16 with no remainder, so 32 is even.85 ÷ 2 = 42 remainder 1, so 85 is odd.
Only the ones digit matters: 50 204 and 76 000 are even; 16 783 and 90 909 are odd.
List the odd numbers from 6 020 to 6 030.
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6 021, 6 023, 6 025, 6 027, 6 029 (five numbers).
1.3Estimating
To estimate is to make a sensible guess of an amount. Use a known amount as your guide, and use words such as "less than", "more than", "about" and "nearly".
Worked example: three pools
Pool R holds 4 000 ℓ. Pool S is about 4 times as high as R, so it holds about 4 × 4 000 ℓ = 16 000 ℓ. Pool T is about twice as high, so it holds about 2 × 4 000 ℓ = 8 000 ℓ.Estimating also checks an answer: if an exact calculation is nowhere near your estimate, look for a mistake.
1.4Rounding off
12 295 rounded off to the nearest thousand is 12 000.
Worked example: round off 38 704 to the nearest ten thousand
Step 1. Look at the digit to the right of the ten thousands: the thousands digit, 8.Step 2. If it is 5, 6, 7, 8 or 9, add 1 to the ten thousands digit: 3 becomes 4.
Step 3. Replace the other digits with 0: 40 000.
| Round off to the nearest … | Look at this digit | Example: 46 195 |
|---|---|---|
| ten | ones | 46 200 |
| hundred | tens | 46 200 |
| thousand | hundreds | 46 000 |
| ten thousand | thousands | 50 000 |
A number exactly in the middle is rounded off to the larger number. 19 500 mℓ becomes 20 000 mℓ to the nearest thousand.
All numbers from 45 000 to 54 999 become 50 000 when rounded off to the nearest ten thousand.
Round off 79 638 to the nearest hundred, thousand and ten thousand.
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Hundred: 79 600 (tens digit 3). Thousand: 80 000 (hundreds digit 6). Ten thousand: 80 000 (thousands digit 9).
1.5Number patterns
A number pattern follows a rule. Find the rule by subtracting two numbers that are next to each other.
Worked example
20 730, 30 730, 40 730, 50 730, 60 730: only the ten thousands digit changes, so the pattern increases by ten thousands.88 275, 87 275, ___, 85 275: each number is 1 000 less, so the missing number is 87 275 − 1 000 = 86 275.
State the number pattern: 26 709, 26 705, 26 701, 26 697, 26 693.
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26 709 − 26 705 = 4, so the pattern decreases by fours.
1.6Addition, subtraction, multiplication and division
Addition
Line up the digits by place value and start from the ones. Regroup when a place has 10 or more.
Worked example: 18 045 + 7 237
5 ones + 7 ones = 12 ones: write 2, regroup 1 ten.4 tens + 3 tens + 1 ten = 8 tens.
0 hundreds + 2 hundreds = 2 hundreds.
8 thousands + 7 thousands = 15 thousands: write 5, regroup 1 ten thousand.
1 + 1 = 2 ten thousands. Answer: 25 282.
Estimate first. Round each number to the nearest thousand: 18 000 + 7 000 = 25 000. The exact answer 25 282 is near 25 000, so it is reasonable.
Subtraction
Worked example: 26 467 − 1 295
Line up the digits. Ones: 7 − 5 = 2. Tens: 6 − 9 is not possible, so take 1 hundred (10 tens): 16 − 9 = 7. Hundreds: 3 − 2 = 1. Thousands: 6 − 1 = 5. Ten thousands: 2.26 467 − 1 295 = 25 172. Check with addition: 25 172 + 1 295 = 26 467.
Keep the digits in the right columns. In 26 467 − 1 117 the 1 117 must be lined up under 6 467, not under 26 46.
Calculate 60 000 − 7 085.
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Take 1 from both numbers: 59 999 − 7 084 = 52 915. (The difference stays the same.)
Multiplication
| Multiply by | Rule | Example |
|---|---|---|
| 10 | write one 0 at the end | 12 × 10 = 120 |
| 100 | write two 0s at the end | 420 × 100 = 42 000 |
| 1 000 | write three 0s at the end | 65 × 1 000 = 65 000 |
Worked example: 49 × 76 (two-digit multiplier)
Step 1. Multiply by the ones: 49 × 6 = 294.Step 2. Multiply by the tens: 49 × 70 = 3 430.
Step 3. Add the two products: 294 + 3 430 = 3 724.
Estimate to check: 49 × 76 is about 50 × 80 = 4 000, and 3 724 is close to that.
When multiplying by the tens digit, put a 0 in the ones place first. Forgetting it makes the answer ten times too small.
Calculate 38 × 20, then 306 × 1 000.
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38 × 20 = 38 × 2 × 10 = 76 × 10 = 760. 306 × 1 000 = 306 000.
Division
Divide by 10, 100 or 1 000 by taking off 1, 2 or 3 zeros: 12 400 ÷ 10 = 1 240; 59 300 ÷ 100 = 593; 43 000 ÷ 1 000 = 43. Division is the inverse of multiplication, so 70 600 ÷ 100 = 706 because 706 × 100 = 70 600.
Worked example: 2 592 ÷ 12
12 into 25 goes 2 times (24), remainder 1. Bring down 9: 19. 12 goes into 19 once (12), remainder 7. Bring down 2: 72. 12 goes into 72 six times.2 592 ÷ 12 = 216. Check: 216 × 12 = 2 592.
Division with a remainder: 14 186 ÷ 32
14 186 ÷ 32 = 443 remainder 10. Check: 443 × 32 = 14 176, and 14 176 + 10 = 14 186.Divide 9 282 by 34.
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273. Check: 273 × 34 = 9 282.
1.7Mixed operations
Mixed addition and subtraction: 24 336 + 1 560 − 4 120
Work from left to right. 24 336 + 1 560 = 25 896; 25 896 − 4 120 = 21 776.Mixed multiplication and division: 630 × 15 ÷ 30
Work from left to right: 630 × 15 = 9 450, then 9 450 ÷ 30 = 315.Shortcut: 630 ÷ 30 = 21 first, then 21 × 15 = 315. Same answer, easier sums.
1.8Finding an unknown
Use the opposite operation
18 + n = 30, so n = 30 − 18 = 12.k − 56 = 28, so k = 28 + 56 = 84.
92 − y = 73, so y = 92 − 73 = 19.
Check by putting the value back: 84 − 56 = 28.
1.9Solving problems
Follow four steps: understand the problem, plan a strategy (a diagram, a table, a number sentence), solve, then check.
Worked example: mangrove seedlings
27 600 seedlings were planted in districts R, S and T. R had 17 930 and S had 8 752. How many in T?Number sentence: 27 600 − 17 930 − 8 752 = 9 670 − 8 752 = 918.
Check: 918 + 8 752 + 17 930 = 27 600.
A school coop has 21 680 items. It has 715 badges and 4 030 magnets; the rest are flags. How many flags?
Show answerHide answer
715 + 4 030 = 4 745. 21 680 − 4 745 = 16 935 flags.
Summary
- Numbers up to 100 000: read in groups; a 0 is not said but is written.
- Place value is the position; digit value is what the digit is worth (6 in 62 307 is 60 000).
- Compare from the left; ascending is smallest first, descending is largest first.
- Even numbers end in 0, 2, 4, 6, 8; odd numbers end in 1, 3, 5, 7, 9.
- Rounding off: look at the digit to the right of the place; 5 or more rounds up, and the middle value goes up.
- Find a number pattern's rule by subtracting neighbours.
- Add and subtract by place value; estimate to check; subtraction is checked by addition.
- To multiply by 10, 100, 1 000 add 1, 2, 3 zeros; to divide, take them off.
- Division is checked by multiplication (then add the remainder).
- For unknowns use the opposite operation. For problems: understand, plan, solve, check.