Mathematics notes, unit 1
Whole Numbers and Operations
Textbook Mathematics Year 5 (DLP), Part 1 and Part 2, pages Part 1, 1–64
What this chapter covers
- You can read and write numbers up to 1 000 000 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 identify prime numbers (1.2)
- You can complete and state number patterns (1.5)
- You can estimate quantities and round off numbers to the nearest thousand, ten thousand and hundred thousand (1.3, 1.4)
- You can add, subtract, multiply and divide whole numbers up to 1 000 000 (1.6)
- You can solve mixed operations with brackets, find unknown numbers and solve problems (1.7, 1.8, 1.9)
1.1Reading and writing numbers up to 1 000 000
In Year 5 you work with six-digit numbers, all the way up to 1 000 000 (one million). You meet them in real data: the number of passengers at Langkawi International Airport in the first quarter of 2018 was 602 337.
Six places, from the left: hundred thousands, ten thousands, thousands, hundreds, tens, ones. Say the thousands group first (the first three digits), then the last three digits.
| Numeral | In words |
|---|---|
| 602 337 | six hundred two thousand three hundred and thirty-seven |
| 200 339 | two hundred thousand three hundred and thirty-nine |
| 125 070 | one hundred twenty-five thousand and seventy |
| 542 000 | five hundred forty-two thousand |
| 900 017 | nine hundred thousand and seventeen |
| 1 000 000 | one million |
Do not read a six-digit number digit by digit. 200 339 is not "two hundred three three nine". And zeros are written in the numeral but not said: "nine hundred thousand and seventeen" is 900 017, not 9 017 or 90 017.
Say and only before the last two digits (tens and ones) when they are less than 100: 125 070 is "...thousand and seventy" and 602 337 is "...three hundred and thirty-seven". You may write "hundred" or "one hundred" for 100.
Write "four hundred six thousand and thirty-five" in numerals. Then write 770 835 in words.
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406 035. 770 835 is seven hundred seventy thousand eight hundred and thirty-five.
1.2Place value, digit value and partitioning
Every digit sits in a place. Place value is the name of the place. Digit value is how much the digit is worth in that place.
Worked example: partition 781 436 and 531 089
Step 1. By place value: 781 436 = 7 hundred thousands + 8 ten thousands + 1 thousand + 4 hundreds + 3 tens + 6 ones.Step 2. By digit value: 781 436 = 700 000 + 80 000 + 1 000 + 400 + 30 + 6.
Step 3. 531 089 has a 0 in the hundreds place. By place value you must write it: 5 hundred thousands + 3 ten thousands + 1 thousand + 0 hundreds + 8 tens + 9 ones.
Step 4. By digit value you may leave the 0 out: 531 089 = 500 000 + 30 000 + 1 000 + 80 + 9.
The parts can be in any order, but each part must match its place. 400 + 600 000 + 7 + 30 + 10 000 is still 610 437. Always write the number back in order: hundred thousands first.
What are the place value and digit value of the underlined digit in 308 715? Then complete: 340 935 = ______ + 40 000 + 900 + 30 + 5.
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The digit is 0: place value ten thousands, digit value 0. 340 935 = 300 000 + 40 000 + 900 + 30 + 5.
1.3Comparing and arranging numbers
A toy shop made four models: robot 118 200 blocks, building 320 915, house 120 290 and car 98 050 blocks.
How to compare two numbers
Step 1. Count the digits. A six-digit number is larger than a five-digit number, so 118 200 > 98 050.Step 2. Same number of digits? Compare from the left: hundred thousands first.
Step 3. Equal? Compare the ten thousands, then thousands, hundreds, tens and ones.
Step 4. 118 200 and 120 290 both have 1 hundred thousand. Ten thousands: 2 is more than 1, so 120 290 > 118 200.
Ascending order: from the smallest to the largest.
Descending order: from the largest to the smallest.
Descending order: from the largest to the smallest.
Ascending order of the four models: 98 050, 118 200, 120 290, 320 915. On a number line the larger number is always further to the right.
A list in ascending or descending order does not always form a pattern. The gaps between the numbers can all be different.
Possible values: 600 128, 600 131, T, 600 134, 600 137
T must be more than 600 131 and less than 600 134. So T can be 600 132 or 600 133.Which is larger, 396 542 or 395 642? Arrange 240 121, 240 212, 240 112 and 240 122 in ascending order.
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The digits are the same up to the hundred thousands and ten thousands. Compare the thousands: 6 is more than 5, so 396 542 is larger. Ascending: 240 112, 240 121, 240 122, 240 212.
1.4Prime numbers
A prime number is a number that is greater than 1 and can only be divided by 1 and itself.
2, 3 and 5 are prime. 4 can also be divided by 2, and 6 can be divided by 2 and 3, so 4 and 6 are not prime. 1 is not prime either, because it is not greater than 1.
Is 51 prime? Is 53 prime?
Step 1. Look for a factor other than 1 and itself.Step 2. 5 + 1 = 6, so 51 can be divided by 3: 51 = 3 × 17. It is not prime.
Step 3. 53 cannot be divided by 2, 3, 5 or 7. It is prime.
2 is the only even prime number. Every other even number can be divided by 2, so it is not prime. Odd numbers are not always prime: 9, 15 and 51 are not.
List the prime numbers from 20 to 40. Why is 30 not prime?
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23, 29, 31, 37. 30 is not prime because it can be divided by other numbers, such as 2, 3, 5 and 10.
1.5Number patterns
In a number pattern the same amount is added or subtracted each time. Say whether it is ascending (increasing) or descending (decreasing), and state the amount.
Find the pattern by looking at the digits that change
503 405, 603 405, 703 405, 803 405: only the hundred thousands digit changes, by 1 each time. The pattern is ascending by hundred thousands (+100 000).219 984, 219 980, 219 976, 219 972: each number is 4 less. The pattern is descending by fours (−4).
127 450, 126 450, 125 450, 124 450: the thousands digit goes down by 1. Descending by thousands (−1 000).
To find the rule, subtract two numbers that sit next to each other, then check it with the next pair.
Complete and state the pattern: 959 289, 859 289, ______, 659 289, 559 289.
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The missing number is 759 289. The pattern is descending by hundred thousands (−100 000).
1.6Estimating and rounding off
Estimating quantities
To estimate, compare with a quantity you know. Tank A is full with 200 000 litres. Tank B holds about three quarters as much.
Estimate by using a multiple
Half of the bricks on pallet A weigh about 370 kg. Pallet B has about 2 times as many bricks as pallet A, so we first find the mass of all of pallet A.Step 1. All of pallet A: about 2 × 370 = 740 kg.
Step 2. Pallet B is 2 times pallet A: about 2 × 740 = 1 480 kg.
An estimate is "about" or "approximately". Say more than, less than or about. It is not an exact answer, so a close value is accepted.
Rounding off numbers
You can round off a six-digit number to the nearest thousand, ten thousand or hundred thousand. Look at the digit just to the right of the place you are rounding to.
Digit on the right is 0 to 4: the digit stays the same. Digit on the right is 5 to 9: add 1 to the digit. In both cases, all the digits to the right become 0.
Worked examples
241 494 to the nearest ten thousand. The thousands digit is 1 (0 to 4), so the 4 stays. Answer: 240 000.264 405 to the nearest hundred thousand. The ten thousands digit is 6 (5 to 9), so 2 becomes 3. Answer: 300 000.
RM380 590 to the nearest hundred thousand ringgit. The ten thousands digit is 8, so 3 becomes 4. Answer: RM400 000.
| Round off to the nearest … | Look at this digit | Example: 573 862 |
|---|---|---|
| thousand | hundreds | 574 000 |
| ten thousand | thousands | 570 000 |
| hundred thousand | ten thousands | 600 000 |
Round off in one step, from the original number. Do not round to the thousand first and then round that answer again.
Round off 129 475 to the nearest thousand, ten thousand and hundred thousand.
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Thousand: hundreds digit 4, so 129 000. Ten thousand: thousands digit 9, so 130 000. Hundred thousand: ten thousands digit 2, so 100 000.
1.7Addition and subtraction
Addition
Line up the digits by place value. Add from the ones. Regroup when a column makes 10 or more.
182 587 + 237 317
Step 1. Ones: 7 + 7 = 14 ones. Write 4, regroup 1 ten.Step 2. Tens: 8 + 1 + 1 = 10 tens. Write 0, regroup 1 hundred.
Step 3. Hundreds: 5 + 3 + 1 = 9. Thousands: 2 + 7 = 9.
Step 4. Ten thousands: 8 + 3 = 11. Write 1, regroup 1 hundred thousand. Hundred thousands: 1 + 2 + 1 = 4.
Answer: 182 587 + 237 317 = 419 904.
Adding many numbers. Start with digits that make 10, and add two numbers at a time. To check a long sum, estimate first: 745 + 1 903 + 35 298 + 895 575 is about 700 + 1 900 + 35 300 + 895 600 = 933 500. The exact answer is 933 521, so it is reasonable.
Unknown in an addition. Use subtraction. If □ + 4 598 = 195 800, then □ = 195 800 − 4 598 = 191 202.
Add 980 + 2 350 + 145 920.
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980 + 2 350 = 3 330, and 3 330 + 145 920 = 149 250.
Subtraction
Line up the digits. Subtract from the ones. If the top digit is smaller, regroup from the next place to the left.
100 685 − 5 322 (difference in paddy area, hectares)
Step 1. Ones 5 − 2 = 3. Tens 8 − 2 = 6. Hundreds 6 − 3 = 3.Step 2. Thousands: 0 − 5 cannot be done. Regroup from the hundred thousands: 1 hundred thousand = 9 ten thousands + 10 thousands, so 10 − 5 = 5.
Step 3. Ten thousands: 9 − 0 = 9. Hundred thousands: 0.
Answer: 100 685 − 5 322 = 95 363.
Two subtractions: 600 000 − 24 750 − 6 098
Subtract the first two numbers, then subtract the next: 600 000 − 24 750 = 575 250, then 575 250 − 6 098 = 569 152.Shortcut for 600 000 − 24 750: take 1 from both numbers. 599 999 − 24 749 gives the same difference and needs no regrouping.
Unknown in a subtraction. Relate it to addition. 509 108 − □ = 417 293, so □ = 509 108 − 417 293 = 91 815. And □ − 230 491 = 190 357, so □ = 190 357 + 230 491 = 420 848.
Subtract: 305 608 − 43 195 − 178.
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305 608 − 43 195 = 262 413, and 262 413 − 178 = 262 235.
1.8Multiplication and division
Multiplication
By 10, 100 and 1 000
Each time, the digits move one place to the left. Write 1, 2 or 3 zeros after the number.
| Calculation | Answer |
|---|---|
| 35 215 × 10 | 352 150 |
| 8 493 × 100 | 849 300 |
| 764 × 1 000 | 764 000 |
By a 1-digit or 2-digit number
2 604 × 43
Step 1. Multiply by the ones: 2 604 × 3 = 7 812.Step 2. Multiply by the tens: 2 604 × 40 = 104 160 (write a 0 in the ones place first).
Step 3. Add the two products: 7 812 + 104 160 = 111 972.
Estimate first. 7 815 × 21 is about 8 000 × 20 = 160 000. The exact answer is 164 115. It is close to 160 000, so it is reasonable. You can also multiply by 21 as 20 and 1, then add: 156 300 + 7 815.
When you multiply by the tens digit, do not forget the 0 in the ones place of the second line. Without it you get 10 times too small.
Calculate 12 640 × 10, 6 × 70 900 and 14 095 × 70.
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126 400, 425 400 and 986 650 (14 095 × 7 = 98 665, then write one 0).
Division
Division is sharing equally, or repeated subtraction. Start from the left, with the biggest place.
By 10, 100 and 1 000
Take away 1, 2 or 3 zeros: 165 090 ÷ 10 = 16 509; 734 000 ÷ 100 = 7 340; 802 000 ÷ 1 000 = 802. This shortcut is only for numbers that end in enough zeros. When there is a remainder you must do the long division: 632 590 ÷ 100 = 6 325 remainder 90.
871 976 ÷ 56 (build the 56 times table)
56, 112, 168, 224, 280, 336, 392, 448, 504.Step 1. 87 ÷ 56 = 1, left 31. Bring down 1: 311 ÷ 56 = 5 (280), left 31. Bring down 9: 319 ÷ 56 = 5 (280), left 39.
Step 2. Bring down 7: 397 ÷ 56 = 7 (392), left 5. Bring down 6: 56 ÷ 56 = 1.
Answer: 871 976 ÷ 56 = 15 571.
With a remainder: 459 613 ÷ 38
459 613 ÷ 38 = 12 095 remainder 3.Check by multiplying: 12 095 × 38 = 459 610, and 459 610 + 3 = 459 613.
The remainder must be less than the divisor. 124 010 ÷ 1 000 = 124 remainder 10, not "124 remainder 1".
Calculate 205 184 ÷ 4 and 126 400 ÷ 10.
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51 296 and 12 640.
1.9Unknown numbers, mixed operations and problems
Unknown numbers
Use a letter or a box for the number you do not know. Then use the opposite operation.
| Number sentence | Opposite operation | Answer |
|---|---|---|
| k × 12 = 36 | k = 36 ÷ 12 | k = 3 |
| 15 × p = 60 000 | p = 60 000 ÷ 15 | p = 4 000 |
| 1 800 ÷ m = 18 | m = 1 800 ÷ 18 | m = 100 |
| s ÷ 24 = 3 190 | s = 3 190 × 24 | s = 76 560 |
Check by putting your answer back into the sentence: 15 × 4 000 = 60 000. If you are stuck, try the same sentence with small numbers first (2 × 3 = 6, so 2 = 6 ÷ 3).
Find h in h × 7 = 105 and w in w ÷ 1 000 = 7.
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h = 105 ÷ 7 = 15. w = 7 × 1 000 = 7 000.
Mixed operations and brackets
Order of operations
1. Work inside the brackets first.
2. Then multiply or divide, from left to right.
3. Last, add or subtract, from left to right.
1. Work inside the brackets first.
2. Then multiply or divide, from left to right.
3. Last, add or subtract, from left to right.
Worked examples
4 + 3 × 5: multiply first, 3 × 5 = 15, then 4 + 15 = 19. (Adding first gives 35, which is wrong.)3 × (150 + 120): brackets first, 270, then 3 × 270 = 810.
48 − 2 × 4: 2 × 4 = 8, then 48 − 8 = 40.
2 + 12 ÷ 3: 12 ÷ 3 = 4, then 2 + 4 = 6.
RM100 ÷ 4 − RM5: RM25 − RM5 = RM20.
(4 594 − 178) ÷ (47 − 23) = 4 416 ÷ 24 = 184.
Do not just work from left to right. 12 + 7 × 6 is 12 + 42 = 54, not 19 × 6 = 114. And 40 + 18 ÷ 2 is 40 + 9 = 49, not 29.
Calculate 500 − 6 × 45 and (350 + 250) ÷ 12.
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500 − 270 = 230. 600 ÷ 12 = 50.
Solving problems
Follow four steps: understand the problem (underline the important information), plan the strategy (draw a diagram or make a table), solve it, then check your answer (with the opposite operation, or an estimate).
Draw a diagram: greeting cards
Teacher's Day cards: 104 500. Father's Day: 3 500 more than Teacher's Day. Mother's Day: 12 600 more than Father's Day. How many Mother's Day cards?
Choose the number sentence: bread for July
A bakery supplies 3 480 loaves a day. July has 31 days.Given: 3 480 loaves a day. Asked for: loaves in 31 days. Number sentence: 31 × 3 480 = 107 880 loaves.
Check: 10 days is 34 800, so 31 days is 3 × 34 800 + 3 480 = 107 880.
Unknown in a problem: the camp
42 participants register each day. A total of 252 registered in p days. p × 42 = 252, so p = 252 ÷ 42 = 6 days.A factory makes 132 780 fishballs a day. How many in 4 days? If one day's fishballs are packed 20 to a packet, how many packets?
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4 × 132 780 = 531 120 fishballs. 132 780 ÷ 20 = 6 639 packets.
Summary
- Read six-digit numbers from the left: the thousands group, then the last three digits. A 0 is written but not said.
- Place value names the place; digit value is what the digit is worth (the 7 in 781 436 is worth 700 000).
- To compare, count the digits, then compare from the hundred thousands. Ascending = smallest first; descending = largest first.
- A prime number is greater than 1 and can only be divided by 1 and itself. 2 is the only even prime.
- Find a number pattern by subtracting neighbours: ascending or descending by the same amount.
- Rounding off: look at the digit to the right. 0 to 4 stays, 5 to 9 goes up; the digits after become 0.
- Add and subtract in columns by place value; estimate first, then check.
- Multiply by 10, 100, 1 000 by adding zeros; divide by removing zeros (only when there is no remainder). The remainder is always less than the divisor.
- Unknowns: use the opposite operation, then check.
- Order of operations: brackets, then multiply or divide, then add or subtract.
- Problems: understand, plan, solve, check.