Year 6: Place value, negative numbers, rounding, factors and primes

Key words. The value of a digit is the digit times its place value. In 4,250, the 2 is in the hundreds place, so its value is 200.

Method. Find the column of the digit, then multiply the digit by the value of that column.

1. Write the number in columns. Each digit sits in a column, with the ones on the right.
   2,450,000
2. Find the hundred thousands column. Count from the right: ones, tens, hundreds, and so on.
   2,450,000: the hundred thousands column has the digit 4
3. Multiply the digit by the place value. The value of a digit is the digit times the value of its column.
   4 × 100,000 = 400,000

Check. The other digits keep their own values, so this digit is worth 400,000 and no more. ✓

Watch out. The digit is worth its column value, not just the digit. A 6 in the thousands is worth 6,000.

Key words. The value of a digit is the digit times its place value. In 4,250, the 2 is in the hundreds place, so its value is 200.

Method. Find the column of the digit, then multiply the digit by the value of that column.

1. Write the number in columns. Each digit sits in a column, with the ones on the right.
   47,305
2. Find the thousands column. Count from the right: ones, tens, hundreds, and so on.
   47,305: the thousands column has the digit 7
3. Multiply the digit by the place value. The value of a digit is the digit times the value of its column.
   7 × 1,000 = 7,000

Check. The other digits keep their own values, so this digit is worth 7,000 and no more. ✓

Watch out. The digit is worth its column value, not just the digit. A 6 in the thousands is worth 6,000.

Key words. To round, look at the digit just after the place you are rounding to. If it is 5 or more, round up. If it is less than 5, round down.

Method. Look at the digit after the hundreds place, then keep the digits up to that place and make the rest zeros.

1. Look at the digit after the hundreds place. That digit decides whether to round up or down.
   the digit is 5
2. Decide whether to round up or down. 5 is 5 or more, so round up.
   round up
3. Write the answer. Keep the digits before the hundreds place and write 0 in the places after it.
   6,900

Check. 6,900 is a multiple of 100, and it is the closest one to 6,850. ✓

Watch out. Do not just cut the digits off. Look at the digit after the hundreds place first.

Key words. To round, look at the digit just after the place you are rounding to. If it is 5 or more, round up. If it is less than 5, round down.

Method. Look at the digit after the ten thousands place, then keep the digits up to that place and make the rest zeros.

1. Look at the digit after the ten thousands place. That digit decides whether to round up or down.
   the digit is 4
2. Decide whether to round up or down. 4 is less than 5, so round down.
   round down
3. Write the answer. Keep the digits before the ten thousands place and write 0 in the places after it.
   250,000

Check. 250,000 is a multiple of 10,000, and it is the closest one to 254,689. ✓

Watch out. Do not just cut the digits off. Look at the digit after the ten thousands place first.

Key words. A negative number is less than zero. On a number line, numbers get smaller as you move left. The interval between two numbers is the number of steps from one to the other.

Method. Start with the most negative number, then go up the number line one number at a time.

1. Find the smallest number. Negative numbers are smaller than zero, and the bigger the number after the minus sign, the smaller the number.
   -10 is the smallest
2. Keep choosing the next smallest. Work along the list until every number has been chosen.
   -10, -4, 0, 3, 7
3. Write the answer. The numbers are now in order from smallest to biggest.
   -10, -4, 0, 3, 7

Check. Check on a number line: each number is to the right of the one before. ✓

Watch out. -10 is smaller than -4, even though 10 is bigger than 4.

Key words. A negative number is less than zero. On a number line, numbers get smaller as you move left. The interval between two numbers is the number of steps from one to the other.

Method. Start at -3 and move 8 steps down the number line, which is to the left.

1. Start at the first temperature. The first temperature is -3 degrees.
   -3 °C
2. Move down 8 degrees. Falling means going to the left, so take 8 away.
   -3 − 8 = -11
3. Write the answer. The unit is degrees Celsius, as in the question.
   -11

Check. Going back up 8 degrees from -11 gives -3 again. ✓

Watch out. A fall in temperature makes the number smaller, even when it is negative.

Key words. A negative number is less than zero. On a number line, numbers get smaller as you move left. The interval between two numbers is the number of steps from one to the other.

Method. Count the steps from the negative number up to zero, then from zero up to the positive number.

1. Count up from -12 to 0. Each step up the number line adds 1.
   12 steps
2. Count up from 0 to 5. Keep counting up, one step at a time.
   5 steps
3. Add the two parts of the interval. The whole interval is the steps to zero plus the steps from zero.
   12 + 5 = 17

Check. Going the other way: from 5 down to 0 is 5 steps, and from 0 down to -12 is 12 steps, so 17 altogether. ✓

Watch out. Do not subtract the two numbers. Crossing zero means you add the two distances.

Key words. A factor divides exactly into a number. A multiple is in the times table of a number. A prime number has exactly two factors, 1 and itself. The highest common factor (HCF) is the biggest factor two numbers share, and the lowest common multiple (LCM) is the smallest multiple they share.

Method. List the factors of both numbers, then pick the biggest one that is on both lists.

1. List the factors of the first number. A factor divides exactly into the number.
   factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
2. List the factors of the second number. Look for the factors that both lists share.
   factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
3. Find the biggest common factor. The highest common factor is the biggest number on both lists.
   common factors: 1, 2, 3, 4, 6, 12
4. Write the answer.
   12

Check. 24 ÷ 12 = 2 and 36 ÷ 12 = 3, both whole numbers. ✓

Watch out. Check that you have all the factors. Pairs of factors are easy to miss, so list them in order.

Key words. A factor divides exactly into a number. A multiple is in the times table of a number. A prime number has exactly two factors, 1 and itself. The highest common factor (HCF) is the biggest factor two numbers share, and the lowest common multiple (LCM) is the smallest multiple they share.

Method. List the multiples of each number, and find the smallest one that is on both lists.

1. List the multiples of 6. Multiply the number by 1, 2, 3 and so on.
   6, 12, 18, 24, 30, 36, 42, 48 ...
2. List the multiples of 8. Do the same for the other number.
   8, 16, 24, 32, 40, 48 ...
3. Find the smallest multiple on both lists. The first number that appears on both lists is the lowest common multiple.
   smallest common multiple = 24

Check. 24 ÷ 6 = 4 and 24 ÷ 8 = 3, so 24 is a multiple of both. ✓

Watch out. Do not stop at a number that is only on one list. Check both lists before you choose.

Key words. A factor divides exactly into a number. A multiple is in the times table of a number. A prime number has exactly two factors, 1 and itself. The highest common factor (HCF) is the biggest factor two numbers share, and the lowest common multiple (LCM) is the smallest multiple they share.

Method. A prime number has exactly two factors, 1 and itself. Test the primes up to the square root.

1. Look for factors up to the square root. The square root of 91 is about 9.5, so only small primes need testing.
   2, 3, 5, 7
2. Test each prime. Divide by each prime. If it goes in exactly, the number is not prime.
   2 does not divide 91; 3 does not divide 91; 5 does not divide 91; 91 ÷ 7 = 13
3. Write the answer. 7 divides into 91 exactly, so 91 has more than two factors.
   No

Check. The factors of 91 are 1, 7, 13, 91. ✓

Watch out. 1 is not a prime number, and 2 is the only even prime.

Key words. A factor divides exactly into a number. A multiple is in the times table of a number. A prime number has exactly two factors, 1 and itself. The highest common factor (HCF) is the biggest factor two numbers share, and the lowest common multiple (LCM) is the smallest multiple they share.

Method. A prime number has exactly two factors, 1 and itself. Test the primes up to the square root.

1. Look for factors up to the square root. The square root of 97 is about 9.8, so only small primes need testing.
   2, 3, 5, 7
2. Test each prime. Divide by each prime. If it goes in exactly, the number is not prime.
   2 does not divide 97; 3 does not divide 97; 5 does not divide 97; 7 does not divide 97
3. Write the answer. No prime up to 97 divides it, so 97 has only the factors 1 and 97.
   Yes

Check. The factors of 97 are 1, 97. ✓

Watch out. 1 is not a prime number, and 2 is the only even prime.

Key words. A factor divides exactly into a number. A multiple is in the times table of a number. A prime number has exactly two factors, 1 and itself. The highest common factor (HCF) is the biggest factor two numbers share, and the lowest common multiple (LCM) is the smallest multiple they share.

Method. Divide by the smallest prime that goes in, and keep going until you reach a prime. Write every prime as the answer.

1. Divide 60 by 2. Keep dividing by the smallest prime that goes in, until the answer is 1.
   60 ÷ 2 = 30
2. Divide 30 by 2. Keep dividing by the smallest prime that goes in, until the answer is 1.
   30 ÷ 2 = 15
3. Divide 15 by 3. Keep dividing by the smallest prime that goes in, until the answer is 1.
   15 ÷ 3 = 5
4. Divide 5 by 5. Keep dividing by the smallest prime that goes in, until the answer is 1.
   5 ÷ 5 = 1
5. Write the product. Write every prime you used, multiplied together.
   2 × 2 × 3 × 5

Check. 2 × 2 × 3 × 5 = 60. ✓

Watch out. Start with 2, then 3, then 5. Do not write a number that is not prime.

Key words. A factor divides exactly into a number. A multiple is in the times table of a number. A prime number has exactly two factors, 1 and itself. The highest common factor (HCF) is the biggest factor two numbers share, and the lowest common multiple (LCM) is the smallest multiple they share.

Method. List the factors of both numbers, then write down the ones that appear on both lists.

1. List the factors of the first number. Find every number that divides exactly into it.
   factors of 18: 1, 2, 3, 6, 9, 18
2. List the factors of the second number. Do the same for the second number.
   factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
3. Pick out the factors on both lists. A common factor is on both lists.
   common factors: 1, 2, 3, 6
4. Write the answer.
   1, 2, 3, 6

Check. The biggest common factor is 6, which is the highest common factor. ✓

Watch out. Do not forget 1, which is a factor of every number.

Key words. A factor divides exactly into a number. A multiple is in the times table of a number. A prime number has exactly two factors, 1 and itself. The highest common factor (HCF) is the biggest factor two numbers share, and the lowest common multiple (LCM) is the smallest multiple they share.

Method. List the factors of both numbers, then pick the biggest one that is on both lists.

1. List the factors of the first number. A factor divides exactly into the number.
   factors of 45: 1, 3, 5, 9, 15, 45
2. List the factors of the second number. Look for the factors that both lists share.
   factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
3. Find the biggest common factor. The highest common factor is the biggest number on both lists.
   common factors: 1, 3, 5, 15
4. Write the answer.
   15

Check. 45 ÷ 15 = 3 and 60 ÷ 15 = 4, both whole numbers. ✓

Watch out. Check that you have all the factors. Pairs of factors are easy to miss, so list them in order.