Mathematics notes, unit 1
Whole Numbers and Operations
Textbook Mathematics Year 6 SK (DLP, Semakan 2017), pages Part 1, 1–60
What this chapter covers
- You can read, write and partition numbers up to 10 000 000 in numerals and in words (1.1.1)
- You can identify, complete and state number patterns that add, subtract, multiply or divide (1.1.2)
- You can say and write fractions and decimals of a million (1.1.3, 1.1.4)
- You can convert between whole numbers and fractions or decimals of a million (1.1.5)
- You can add, subtract, multiply and divide numbers up to 10 000 000, including fractions and decimals of a million (1.2.1)
- You can solve mixed operations with brackets and find an unknown number (1.2.1)
- You can identify prime numbers and composite numbers (1.3.1)
- You can solve problems with whole numbers and with fractions and decimals of a million (1.4.1)
1.1Reading and writing numbers up to 10 000 000
In Year 6 the numbers grow to millions. The population of a state, the attendance at a stadium or the number of internet users in a country can all be a million or more.
One million is 1 000 000. It is 1 000 thousands. The largest number we use this year is 10 000 000 (ten million), the first number with eight digits. It is 1 more than 9 999 999.
Put the digits into the place value chart, then read from the left. Say the millions, then the thousands, then the hundreds, tens and ones.
| Digit | 6 | 3 | 4 | 0 | 2 | 1 | 5 |
|---|---|---|---|---|---|---|---|
| Place value | millions | hundred thousands | ten thousands | thousands | hundreds | tens | ones |
| Digit value | 6 000 000 | 300 000 | 40 000 | 0 | 200 | 10 | 5 |
So 6 340 215 is six million three hundred and forty thousand two hundred and fifteen.
| Numeral | In words |
|---|---|
| 3 080 506 | three million eighty thousand five hundred and six |
| 7 205 040 | seven million two hundred and five thousand and forty |
| 8 000 080 | eight million and eighty |
| 10 000 000 | ten million |
A digit 0 is never said aloud, but it must be written. "Four million three hundred and two" is 4 000 302, not 4 302 or 4 300 002. Write the number in groups of three digits from the right so you can see which places are empty.
Partitioning
Partition a number by digit value: 6 340 215 = 6 000 000 + 300 000 + 40 000 + 200 + 10 + 5. The 0 in the thousands place is left out.
Write "five million sixty thousand and nine" in numerals. What is the digit value of 3 in 8 035 214?
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5 060 009: 5 millions, 0 hundred thousands, 6 ten thousands, 0 thousands, 0 hundreds, 0 tens, 9 ones. The 3 is in the ten thousands place, so its digit value is 30 000.
1.2Number patterns
A number pattern follows a rule. To find the rule, calculate the difference between two numbers that sit next to each other. Say the rule in words: "in ascending order by eight hundreds".
Worked example: find the missing number
2 140 600, 2 195 600, 2 250 600, x, 2 360 600Step 1. Find the difference: 2 195 600 − 2 140 600 = 55 000.
Step 2. Check another pair: 2 250 600 − 2 195 600 = 55 000. The rule is + 55 000.
Step 3. x = 2 250 600 + 55 000 = 2 305 600.
Step 4. Check: 2 305 600 + 55 000 = 2 360 600. Correct.
Patterns do not always add or subtract. They can also multiply or divide.
- 5, 15, 45, 135, ... multiply by 3 each time. The next number is 135 × 3 = 405.
- 128, 64, 32, 16, ... divide by 2 each time. The next number is 16 ÷ 2 = 8.
- 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9 ... Figure number 7 gives 7 × 7 = 49.
To find where a number sits in a pattern, count how many steps it is from the first number. If the pattern goes up in 6s from 4 095 180, the number 4 095 204 is 24 more, and 24 ÷ 6 = 4 steps later, so it is the fifth number.
Complete 7 657 850, 7 657 450, 7 657 050, ___. State the pattern.
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Each step is 400 less. 7 657 050 − 400 = 7 656 650. The pattern is descending order by four hundreds.
1.3Fraction and decimal of a million
News reports say "2.5 million visitors" or "3¾ million tickets". These are decimals of a million and fractions of a million. To read or say them, say the number and then the word "million": 0.75 million is "zero point seven five million".
To change to a whole number, multiply by 1 000 000. To change a whole number to a decimal of a million, divide by 1 000 000.
| Fraction | Decimal of a million | Whole number |
|---|---|---|
| ½ million | 0.5 million | 500 000 |
| ¼ million | 0.25 million | 250 000 |
| ¾ million | 0.75 million | 750 000 |
| ⅕ million | 0.2 million | 200 000 |
| ⅛ million | 0.125 million | 125 000 |
| 1⁄10 million | 0.1 million | 100 000 |
Worked examples
(a) 5.78 million = 5.78 × 1 000 000 = 5 780 000.(b) 0.674 million = 0.674 × 1 000 000 = 674 000.
(c) 3⅗ million: ⅗ = 0.6, so it is 3.6 million = 3 600 000.
(d) 4 800 000 = 4 800 000 ÷ 1 000 000 = 4.8 million.
(e) 800 000 = 0.8 million = 8⁄10 million = ⅘ million (simplest form).
Fractions of a million must be written in the simplest form. And the zeros matter: 0.07 million is 70 000, not 700 000 and not 7 000.
Write 1 750 000 as a fraction of a million, and 2¼ million as a whole number.
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1 750 000 = 1.75 million = 1¾ million. 2¼ million = 2.25 million = 2 250 000.
1.4Basic operations
Add, subtract, multiply and divide in the usual way. Line up the digits by place value for adding and subtracting. When the question mixes whole numbers, decimals and fractions of a million, change everything to the same form first: all whole numbers, or all decimals of a million.
Worked examples
Add. 2.35 million + 1⅗ million + 85 000Step 1. 1⅗ million = 1.6 million. 85 000 = 0.085 million.
Step 2. 2.350 + 1.600 + 0.085 = 4.035 million = 4 035 000.
Subtract. 7.2 million − 4⅜ million
Step 1. 4⅜ = 4.375, so 4⅜ million = 4.375 million.
Step 2. 7.200 − 4.375 = 2.825 million = 2 825 000.
Multiply. 6 × 0.35 million = 2.10 million = 2 100 000.
Divide. 5½ million ÷ 4
Step 1. 5½ million = 5.5 million.
Step 2. 5.5 ÷ 4 = 1.375, so the answer is 1.375 million = 1 375 000.
Check an answer with the opposite operation. 1.375 million × 4 = 5.5 million, so the division is right. Or estimate first: 5½ ÷ 4 is a little more than 1.
Calculate 4 305 000 − 0.96 million and 0.8 million ÷ 5.
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0.96 million = 960 000, so 4 305 000 − 960 000 = 3 345 000. 0.8 million ÷ 5 = 0.16 million = 160 000.
1.5Mixed operations
Order of operations: (1) work inside the brackets first; (2) then × and ÷; (3) then + and −. With the same level of operation, work from left to right.
Worked examples
(a) 1.2 million − 0.5 million + 0.8 million = 0.7 million + 0.8 million = 1.5 million (left to right).(b) 0.2 million + 9 × 40 = 200 000 + 360 = 200 360 (multiply first).
(c) 4.8 million ÷ 6 − 0.35 million = 0.8 million − 0.35 million = 0.45 million (divide first).
(d) (3.6 million − 1¼ million) × 4 = (3.6 − 1.25) million × 4 = 2.35 million × 4 = 9.4 million (brackets first).
Do not just go left to right. In 200 000 + 9 × 40, adding first gives 200 009 × 40, which is wrong. Multiplication comes before addition.
Calculate (2.5 million + 1½ million) ÷ 8.
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Brackets: 2.5 + 1.5 = 4 million. Then 4 million ÷ 8 = 0.5 million = 500 000.
1.6Finding the unknown
When a letter stands for a missing number, use the opposite operation to undo the calculation. Undo the last step first.
| Problem | Working | Value |
|---|---|---|
| x + 0.6 million = 1.3 million | x = 1.3 million − 0.6 million | 0.7 million |
| m ÷ 5 = 0.08 million | m = 0.08 million × 5 | 0.4 million |
| (k + 0.2 million) × 3 = 1.2 million | k + 0.2 million = 1.2 million ÷ 3 = 0.4 million, so k = 0.4 million − 0.2 million | 0.2 million |
| n − 2 × 0.15 million = 0.45 million | n − 0.3 million = 0.45 million, so n = 0.45 million + 0.3 million | 0.75 million |
Always put the value back in the sentence to check. 0.75 million − 2 × 0.15 million = 0.75 million − 0.3 million = 0.45 million. Correct.
Find the value of p: p ÷ 4 × 3 = 150 000.
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Undo × 3 first: p ÷ 4 = 150 000 ÷ 3 = 50 000. Then p = 50 000 × 4 = 200 000.
1.7Prime numbers and composite numbers
A prime number can be divided exactly only by 1 and itself (2, 3, 5, 7, 11 ...).
A composite number can be divided exactly by 1, itself and at least one other number (4, 9, 51 ...).
1 is neither prime nor composite.
A composite number can be divided exactly by 1, itself and at least one other number (4, 9, 51 ...).
1 is neither prime nor composite.
Worked example: is 51 prime? Is 53 prime?
Step 1. Try dividing 51: 51 ÷ 3 = 17 with no remainder. So 51 can be divided by 1, 3, 17 and 51. It is composite.Step 2. Try 53 by 2, 3, 5 and 7. None divides it exactly. 53 can be divided only by 1 and 53. It is prime.
2 is the only even prime number. Every other even number can be divided by 2, so it is composite. A quick test: a number is divisible by 3 if its digits add up to a multiple of 3 (5 + 1 = 6, so 51 is).
Numbers ending in 3 are not always prime. 33, 63 and 93 are composite.
Which of 57, 59 and 61 are prime?
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57 = 3 × 19 is composite. 59 and 61 are prime.
1.8Solving problems
Follow four steps: understand the problem (underline what is given and what is asked), plan a strategy (draw a bar model, write a number sentence), solve, and check with the opposite operation.
Worked example
Shop A sold 1.35 million tickets. Shop B sold 450 000 more tickets than shop A. How many tickets did the two shops sell altogether?Step 1. Shop B = 1 350 000 + 450 000 = 1 800 000.
Step 2. Total = 1 350 000 + 1 800 000 = 3 150 000 (3.15 million).
Step 3. Check: 3 150 000 − 1 800 000 = 1 350 000. Correct.
Worked example with a number sentence
A factory makes 3.12 million paper cups and packs them equally into 60 cartons. Three cartons go to one cafe. How many cups does the cafe get?Number sentence: 3.12 million ÷ 60 × 3 = 3 120 000 ÷ 60 × 3 = 52 000 × 3 = 156 000 cups.
Look for key words. "More than", "altogether" and "in total" suggest adding. "Left", "remaining" and "difference" suggest subtracting. "Each" and "equally" suggest dividing.
A shop has 0.4 million pens. It sells 6 boxes of 5 000 pens each. How many pens are left?
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0.4 million = 400 000. 6 × 5 000 = 30 000. 400 000 − 30 000 = 370 000 pens.