Year 6 Place Value and Negative Numbers: check-up and SATs-style questions

Key words. The value of a digit depends on its column (its place). Each column is ten times the column on its right. In 4,672,819 the 6 is in the hundred thousands column, so it is worth 6 × 100,000 = 600,000.

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

1. Find the column of the 9. Count the columns from the right. The 9 is in the hundred thousands column.
   5,920,814: the 9 is in the hundred thousands column
2. Multiply the digit by the column value. The value of a digit is the digit times the value of its column.
   9 × 100,000 = 900,000
3. Check the column. Count the columns from the right to be sure.
   Column 6 from the right is the hundred thousands column

Check. Only the digit 9 is worth 900,000. The other digits are worth their own values, so the whole number stays 5,920,814. ✓

Watch out. The 9 is worth 900,000, not 9. Its column decides its value.

Method. Split the number into groups of three digits, then read each group and say its name.

1. Split the number into groups. Each group of three has its own name: million, thousand, or the ones group.
   3,070,405
2. Read each group in words. Say the number in each group, then its name.
   three million | seventy thousand | four hundred and five
3. Join the groups with commas. Use "and" before a last group that is less than 100.
   three million, seventy thousand, four hundred and five

Check. Write the words back as figures: 3,070,405. ✓

Watch out. Zeros still count. 4,000,080 is four million and eighty, not four thousand and eighty.

Method. Find the value of each group, then add the groups together.

1. Find the value of each group. Each name tells you how many zeros the group needs.
   4,000,000 + 60,000 + 12
2. Add the groups. Put them together to make one number.
   4,000,000 + 60,000 + 12 = 4,060,012
3. Check the groups. Read the number back in words to check.
   four million | sixty thousand | twelve

Check. Reading 4,060,012 back gives "four million, sixty thousand and twelve". ✓

Watch out. Put commas after every three digits from the right: 6,209,070.

Key words. To compare two numbers, use the signs < (is less than) and > (is greater than). The open end of the sign faces the bigger number: 9 > 4 and 4 < 9.

Method. Compare the digits from the left. The first place where they differ decides.

1. Line the numbers up and compare from the left. The first column where the digits differ decides which number is bigger.
   7,410,000 and 7,401,999 both have 7 digits.
2. Look at the ten thousands column. The digits are 1 and 0. The larger digit means the larger number.
   1 and 0, so 7,410,000 is bigger
3. Choose the sign. The open end of the sign faces the bigger number.
   7,410,000 > 7,401,999

Check. Compare the other way round: 7,401,999 and 7,410,000 give the opposite sign. ✓

Watch out. Do not compare the last digits first. 4,999,999 is less than 5,000,000, even though it has more 9s.

Key words. To compare two numbers, use the signs < (is less than) and > (is greater than). The open end of the sign faces the bigger number: 9 > 4 and 4 < 9.

Method. Compare the numbers in pairs, then write them from the smallest to the other.

1. Compare the numbers. Look at the number of digits, then compare the digits from the left.
   5,020,000, 502,000, 5,200,000, 5,002,000, 5,020,100
2. Write the numbers from the smallest. Put the smallest number first.
   502,000, 5,002,000, 5,020,000, 5,020,100, 5,200,000
3. Check the order. Each number should be bigger than the one before it.
   502,000 < 5,002,000 < 5,020,000 < 5,020,100 < 5,200,000

Check. The first number is the smallest and the last number is the largest. ✓

Watch out. Do not order by the first digit alone if the numbers have different numbers of digits.

Key words. To round a number to a given place value, find the two multiples of that place value either side of it. Then choose the one the number is nearer to. If the number is exactly halfway, we round up.

Method. Find the two multiples of 100,000 either side of the number, then choose the nearer one.

1. Find the two multiples. The number lies between two multiples of 100,000.
   8,400,000 < 8,465,300 < 8,500,000
2. Check the digit after the rounding digit. Halfway is 50,000. Look at how far the number is past 8,400,000.
   8,465,300 − 8,400,000 = 65,300, and 65,300 is at least 50,000
3. Round up or down. Five or more means round up. Less than five means round down.
   Round up to 8,500,000

Check. 8,500,000 is a multiple of 100,000 and is the closer one to 8,465,300. ✓

Watch out. Look at the digit just to the right of the place you are rounding to. The other digits do not matter for the rule.

Key words. To round a number to a given place value, find the two multiples of that place value either side of it. Then choose the one the number is nearer to. If the number is exactly halfway, we round up.

Method. Find the two multiples of 1,000 either side of the number, then choose the nearer one.

1. Find the two multiples. The number lies between two multiples of 1,000.
   6,043,000 < 6,043,250 < 6,044,000
2. Check the digit after the rounding digit. Halfway is 500. Look at how far the number is past 6,043,000.
   6,043,250 − 6,043,000 = 250, and 250 is less than 500
3. Round up or down. Five or more means round up. Less than five means round down.
   Round down to 6,043,000

Check. 6,043,000 is a multiple of 1,000 and is the closer one to 6,043,250. ✓

Watch out. Look at the digit just to the right of the place you are rounding to. The other digits do not matter for the rule.

Key words. A negative number is a number less than zero. It has a minus sign in front of it, such as −4. Zero is neither positive nor negative. On a number line, numbers get smaller as you move left.

Method. Start at the first number and move 6 steps down. Count through zero if you need to.

1. Start at the first number. Find it on the number line or thermometer.
   Start at −4 °C
2. Move 6 steps down. Count one step at a time. Each step changes the number by 1.
   −4 − 6
3. Write the new number.
   −10 °C

Check. −4 − 6 = −10. Going back: −10 + 6 = −4. ✓

Watch out. Going down past zero makes the number negative. Count through zero, do not stop at zero.

Key words. An interval is the difference between two numbers. To find an interval across zero, count the steps from one number to zero, then from zero to the other number, and add the two parts together.

Method. Find the number of steps between the two numbers. If the numbers are on opposite sides of zero, add the steps to zero.

1. Count the steps from −12 up to zero. Zero is the middle of the number line.
   12 steps
2. Count the steps from zero up to 9. Keep counting the same way, one step at a time.
   9 steps
3. Add the two parts. The total number of steps is the interval.
   12 + 9 = 21

Check. Count on from −12 to 9 one step at a time: you pass zero and count 21 steps in total. ✓

Watch out. Do not subtract two negative numbers in the same way as positive numbers. Count the steps.

Key words. A negative number is a number less than zero. It has a minus sign in front of it, such as −4. Zero is neither positive nor negative. On a number line, numbers get smaller as you move left.

Method. Start at the first number and move 80 steps up. Count through zero if you need to.

1. Start at the first number. Find it on the number line or thermometer.
   Start at −235 m
2. Move 80 steps up. Count one step at a time. Each step changes the number by 1.
   −235 + 80
3. Write the new number.
   −155 m

Check. −235 + 80 = −155. Going back: −155 − 80 = −235. ✓

Watch out. Going down past zero makes the number negative. Count through zero, do not stop at zero.

Key words. The value of a digit depends on its column (its place). Each column is ten times the column on its right. In 4,672,819 the 6 is in the hundred thousands column, so it is worth 6 × 100,000 = 600,000.

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

1. Find the column of the 4. Count the columns from the right. The 4 is in the hundred thousands column.
   1,438,920: the 4 is in the hundred thousands column
2. Multiply the digit by the column value. The value of a digit is the digit times the value of its column.
   4 × 100,000 = 400,000
3. Check the column. Count the columns from the right to be sure.
   Column 6 from the right is the hundred thousands column

Check. Only the digit 4 is worth 400,000. The other digits are worth their own values, so the whole number stays 1,438,920. ✓

Watch out. The 4 is worth 400,000, not 4. Its column decides its value.

Key words. To compare two numbers, use the signs < (is less than) and > (is greater than). The open end of the sign faces the bigger number: 9 > 4 and 4 < 9.

Method. Compare the numbers in pairs, then write them from the largest to the other.

1. Compare the numbers. Look at the number of digits, then compare the digits from the left.
   4,250,000, 4,520,000, 4,205,000, 4,502,000
2. Write the numbers from the largest. Put the largest number first.
   4,520,000, 4,502,000, 4,250,000, 4,205,000
3. Check the order. Each number should be smaller than the one before it.
   4,520,000 > 4,502,000 > 4,250,000 > 4,205,000

Check. The first number is the largest and the last number is the smallest. ✓

Watch out. Do not order by the first digit alone if the numbers have different numbers of digits.

Method. Split the number into groups of three digits, then read each group and say its name.

1. Split the number into groups. Each group of three has its own name: million, thousand, or the ones group.
   6,002,050
2. Read each group in words. Say the number in each group, then its name.
   six million | two thousand | fifty
3. Join the groups with commas. Use "and" before a last group that is less than 100.
   six million, two thousand and fifty

Check. Write the words back as figures: 6,002,050. ✓

Watch out. Zeros still count. 4,000,080 is four million and eighty, not four thousand and eighty.

Method. The first digit cannot be zero. Use the smallest digit that is not zero, then the rest in order from smallest.

1. Choose the first digit. A number cannot start with a zero, so the first digit must be a digit that is not zero.
   The smallest digit that is not zero is 3.
2. Write the other digits from smallest to biggest. The smaller digits go in the columns on the right.
   3 then 0 0 5 6 8 9
3. Write the number.
   3,005,689

Check. The number has 7 digits, and every digit is used once. ✓

Watch out. Zero can be used, but not as the first digit.

Key words. To round a number to a given place value, find the two multiples of that place value either side of it. Then choose the one the number is nearer to. If the number is exactly halfway, we round up.

Method. Find the two multiples of 1,000 either side of the number, then choose the nearer one.

1. Find the two multiples. The number lies between two multiples of 1,000.
   3,749,000 < 3,749,500 < 3,750,000
2. Check the digit after the rounding digit. Halfway is 500. Look at how far the number is past 3,749,000.
   3,749,500 − 3,749,000 = 500, and 500 is at least 500
3. Round up or down. Five or more means round up. Less than five means round down.
   Round up to 3,750,000

Check. 3,750,000 is a multiple of 1,000 and is the closer one to 3,749,500. ✓

Watch out. Look at the digit just to the right of the place you are rounding to. The other digits do not matter for the rule.

Key words. To compare two numbers, use the signs < (is less than) and > (is greater than). The open end of the sign faces the bigger number: 9 > 4 and 4 < 9.

Method. Compare the numbers in pairs, then write them from the smallest to the other.

1. Compare the numbers. Look at the number of digits, then compare the digits from the left.
   2,300,000, 230,000, 2,030,000, 2,003,000
2. Write the numbers from the smallest. Put the smallest number first.
   230,000, 2,003,000, 2,030,000, 2,300,000
3. Check the order. Each number should be bigger than the one before it.
   230,000 < 2,003,000 < 2,030,000 < 2,300,000

Check. The first number is the smallest and the last number is the largest. ✓

Watch out. Do not order by the first digit alone if the numbers have different numbers of digits.

Method. Round each option to the nearest 1,000,000, then choose the one that gives the target.

1. Round every option. Look at the digit after the millions. Five or more rounds up.

   Option	Number	Rounded	Target?
   A	5,250,000	5,250,000 rounds to 5,000,000	no
   B	5,480,000	5,480,000 rounds to 5,000,000	no
   C	6,520,000	6,520,000 rounds to 7,000,000	no
   D	6,490,000	6,490,000 rounds to 6,000,000	yes

2. Choose the answer. Only one option rounds to the target.
   6,490,000 is option D

Check. Check the chosen number: 6,490,000 is 490,000 away from the target. ✓

Watch out. Check every option, not just the first one that looks close.

Key words. An interval is the difference between two numbers. To find an interval across zero, count the steps from one number to zero, then from zero to the other number, and add the two parts together.

Method. Find the number of steps between the two numbers. If the numbers are on opposite sides of zero, add the steps to zero.

1. Count the steps from −7 up to zero. Zero is the middle of the number line.
   7 steps
2. Count the steps from zero up to 5. Keep counting the same way, one step at a time.
   5 steps
3. Add the two parts. The total number of steps is the interval.
   7 + 5 = 12

Check. Count on from −7 to 5 one step at a time: you pass zero and count 12 steps in total. ✓

Watch out. Do not subtract two negative numbers in the same way as positive numbers. Count the steps.

Key words. A negative number is a number less than zero. It has a minus sign in front of it, such as −4. Zero is neither positive nor negative. On a number line, numbers get smaller as you move left.

Method. Compare the two numbers on a number line. The number nearer to zero is bigger when both are negative.

1. Find where each number is. Start at zero and count down.
   −14 is 14 steps below zero. −9 is 9 steps below zero.
2. Compare them. Further below zero means smaller.
   −14 is further below zero, so it is smaller.
3. Explain the mistake. Sam forgot that the bigger minus number is the smaller number.
   −9 is greater than −14.

Check. Count down: −9 is reached first, so it is greater. ✓

Watch out. Compare negative numbers by the number on the number line, not by the size of the digits alone.

Key words. A negative number is a number less than zero. It has a minus sign in front of it, such as −4. Zero is neither positive nor negative. On a number line, numbers get smaller as you move left.

Method. Work out the diver's depth after each move. Use the answer from the first move as the start of the second.

1. After the first move. She rises 135 m from −240 m.
   −240 + 135 = −105
2. After the second move. She dives 60 m from −105 m.
   −105 − 60 = −165
3. Write the answer with its unit.
   She is at −165 m

Check. Going back: −165 + 60 = −105, and −105 − 135 = −240. ✓

Watch out. Dive means go down, so the number gets more negative.