Year 6: Intervals across zero
Part of the Year 6 place value and negative numbers chapter. Chapter overview · whole chapter on one page
Intervals across zero
An interval is the gap between two numbers. On a number line we can count the steps. When the two numbers are on opposite sides of zero, the easiest way is to count the steps from each number to zero, and then add the two parts.
From −6 to 5: there are 6 steps from −6 up to zero, and 5 steps from zero up to 5. The interval is 6 + 5 = 11.
When both numbers are on the same side of zero, you do not cross zero. Subtract the smaller distance from the bigger distance. From −9 to −4 there are 9 steps to zero and 4 steps to zero, so the interval is 9 − 4 = 5.
Key ideas
- An interval is the number of steps between two numbers.
- Numbers on opposite sides of zero: add the steps to zero from each number.
- Numbers on the same side of zero: subtract the smaller distance from the bigger distance.
- A rise in temperature is the interval from the lower temperature up to the higher one.
- Check by counting on from the first number to the second, one step at a time.
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.
- Count the steps from −8 up to zero. Zero is the middle of the number line.8 steps
- Count the steps from zero up to 5. Keep counting the same way, one step at a time.5 steps
- Add the two parts. The total number of steps is the interval.8 + 5 = 13
Check. Count on from −8 to 5 one step at a time: you pass zero and count 13 steps in total. ✓
Watch out. Do not subtract two negative numbers in the same way as positive numbers. Count the steps.
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.
- Find the distance of each number from zero. Both numbers are negative, so use the size of each number without its minus sign.−9 is 9 from zero, and −4 is 4 from zero
- Subtract the smaller distance from the bigger one. Both numbers are on the same side of zero.9 − 4 = 5
- Write the interval. 5
Check. Count up from −9 to −4: that is 5 steps. ✓
Watch out. Do not subtract two negative numbers in the same way as positive numbers. Count the steps.
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.
- Count the steps from −7 up to zero. Zero is the middle of the number line.7 steps
- Count the steps from zero up to 3. Keep counting the same way, one step at a time.3 steps
- Add the two parts. The total number of steps is the interval.7 + 3 = 10
Check. Count on from −7 to 3 one step at a time: you pass zero and count 10 steps in total. ✓
Watch out. Do not subtract two negative numbers in the same way as positive numbers. Count the steps.
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.
- Count the steps from −3 up to zero. Zero is the middle of the number line.3 steps
- Count the steps from zero up to 12. Keep counting the same way, one step at a time.12 steps
- Add the two parts. The total number of steps is the interval.3 + 12 = 15
Check. Count on from −3 to 12 one step at a time: you pass zero and count 15 steps in total. ✓
Watch out. Do not subtract two negative numbers in the same way as positive numbers. Count the steps.
Watch out
Do not subtract when the numbers are on opposite sides of zero. From −4 to 6 is not 6 − 4 = 2. It is 4 + 6 = 10 steps. Also do not forget the step through zero: count it when you cross zero.Exercise 1E Intervals across zero
Non-calculatorKey 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.
- Count the steps from −4 up to zero. Zero is the middle of the number line.4 steps
- Count the steps from zero up to 6. Keep counting the same way, one step at a time.6 steps
- Add the two parts. The total number of steps is the interval.4 + 6 = 10
Check. Count on from −4 to 6 one step at a time: you pass zero and count 10 steps in total. ✓
Watch out. Do not subtract two negative numbers in the same way as positive numbers. Count the steps.
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.
- Count the steps from −7 up to zero. Zero is the middle of the number line.7 steps
- Count the steps from zero up to 2. Keep counting the same way, one step at a time.2 steps
- Add the two parts. The total number of steps is the interval.7 + 2 = 9
Check. Count on from −7 to 2 one step at a time: you pass zero and count 9 steps in total. ✓
Watch out. Do not subtract two negative numbers in the same way as positive numbers. Count the steps.
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.
- Find the distance of each number from zero. Both numbers are negative, so use the size of each number without its minus sign.−12 is 12 from zero, and −5 is 5 from zero
- Subtract the smaller distance from the bigger one. Both numbers are on the same side of zero.12 − 5 = 7
- Write the interval. 7
Check. Count up from −12 to −5: that is 7 steps. ✓
Watch out. Do not subtract two negative numbers in the same way as positive numbers. Count the steps.
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.
- Count the steps from −3 up to zero. Zero is the middle of the number line.3 steps
- Count the steps from zero up to 8. Keep counting the same way, one step at a time.8 steps
- Add the two parts. The total number of steps is the interval.3 + 8 = 11
Check. Count on from −3 to 8 one step at a time: you pass zero and count 11 steps in total. ✓
Watch out. Do not subtract two negative numbers in the same way as positive numbers. Count the steps.
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.
- Count the steps from −10 up to zero. Zero is the middle of the number line.10 steps
- Count the steps from zero up to 4. Keep counting the same way, one step at a time.4 steps
- Add the two parts. The total number of steps is the interval.10 + 4 = 14
Check. Count on from −10 to 4 one step at a time: you pass zero and count 14 steps in total. ✓
Watch out. Do not subtract two negative numbers in the same way as positive numbers. Count the steps.
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.
- Count the steps from −6 up to zero. Zero is the middle of the number line.6 steps
- Count the steps from zero up to 4. Keep counting the same way, one step at a time.4 steps
- Add the two parts. The total number of steps is the interval.6 + 4 = 10
Check. Count on from −6 to 4 one step at a time: you pass zero and count 10 steps in total. ✓
Watch out. Do not subtract two negative numbers in the same way as positive numbers. Count the steps.
| Day | Temperature at noon |
|---|---|
| Monday | −3 °C |
| Tuesday | 5 °C |
| Wednesday | −8 °C |
| Thursday | 2 °C |
| Friday | −1 °C |
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.
- Count the steps from −8 up to zero. Zero is the middle of the number line.8 steps
- Count the steps from zero up to 5. Keep counting the same way, one step at a time.5 steps
- Add the two parts. The total number of steps is the interval.8 + 5 = 13
Check. Count on from −8 to 5 one step at a time: you pass zero and count 13 steps in total. ✓
Watch out. Do not subtract two negative numbers in the same way as positive numbers. Count the steps.
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.
- Count the steps from −5 up to zero. Zero is the middle of the number line.5 steps
- Count the steps from zero up to 3. Keep counting the same way, one step at a time.3 steps
- Add the two parts. The total number of steps is the interval.5 + 3 = 8
Check. Count on from −5 to 3 one step at a time: you pass zero and count 8 steps in total. ✓
Watch out. Do not subtract two negative numbers in the same way as positive numbers. Count the steps.
Method. Count the whole numbers on the line, including both ends.
- Count the negative numbers. These are −4, −3, −2 and −1.4 numbers
- Count zero. Zero is a whole number too.1 number
- Count the positive numbers. These are 1, 2, 3, 4 and 5.5 numbers
- Add them. 4 + 1 + 5 = 10
Check. Count on from −4 to 5: −4, −3, −2, −1, 0, 1, 2, 3, 4, 5. ✓
Watch out. Both ends are included, so the list is not one less than the interval.
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.
- Find the distance of each number from zero. Both numbers are negative, so use the size of each number without its minus sign.−120 is 120 from zero, and −35 is 35 from zero
- Subtract the smaller distance from the bigger one. Both numbers are on the same side of zero.120 − 35 = 85
- Write the interval. 85
Check. Count up from −120 to −35: that is 85 steps. ✓
Watch out. Do not subtract two negative numbers in the same way as positive numbers. Count the steps.
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.
- Count the steps from −85 up to zero. Zero is the middle of the number line.85 steps
- Count the steps from zero up to 140. Keep counting the same way, one step at a time.140 steps
- Add the two parts. The total number of steps is the interval.85 + 140 = 225
Check. Count on from −85 to 140 one step at a time: you pass zero and count 225 steps in total. ✓
Watch out. Do not subtract two negative numbers in the same way as positive numbers. Count the steps.
Method. Work out each interval, then compare them.
- Find the first interval. From −3 to 8 crosses zero, so add the steps.3 + 8 = 11
- Find the second interval. From −10 to −2 is on one side of zero, so subtract.10 − 2 = 8
- Compare. The bigger interval is the first one.11 is more than 8
Check. Count the steps for each: 11 for the first, 8 for the second. ✓
Watch out. Do not compare the two numbers at the end of each interval. Compare the intervals.
Method. Count the steps from −9 to zero, then count the rest of the steps to the positive number.
- Count the steps from −9 to zero. That uses 9 of the 14 steps.9 steps
- Count the steps left. The remaining steps go past zero.14 − 9 = 5
- Write the number. 5
Check. 9 steps to zero and 5 steps up to 5 make 9 + 5 = 14 steps. ✓
Watch out. Count the steps to zero first, then the rest.
Method. Write the two parts of the interval. Compare the sum with the positive number.
- Name the two numbers. Let the size of the negative number (without its minus sign) be a, and the positive number be b.The interval is a + b
- Compare with b. Since a is positive, a + b is bigger than b.a + b > b
- Check the same for a. a + b is also bigger than a, because b is positive.a + b > a
Check. Example: from −3 to 4 is 3 + 4 = 7, and 7 is bigger than 4. ✓
Watch out. Always needs a reason that works for every case.
Method. Use the two clues. Write the positive number as the negative size plus 3, and use the total of 17.
- Write the clues as numbers. Let the size of the negative number be a. The positive number is a + 3.a + (a + 3) = 17
- Solve the equation. Two lots of a plus 3 make 17, so take 3 away and halve the rest.2a + 3 = 17, so 2a = 14, so a = 7
- Find the two numbers. The negative number is −7, and the positive number is 7 + 3 = 10.−7 and 10
Check. Check: from −7 to 10 is 7 + 10 = 17 steps. The positive number is 3 more than 7. ✓
Watch out. Check both clues before you answer.