Year 5: Estimating and checking
Part of the Year 5 long multiplication chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Estimating and checking
An estimate is an answer that is close, but not exact. To estimate a multiplication, round each number to one digit followed by zeros, then multiply the rounded numbers. The number line shows 347 is closer to 300 than to 400, so we round it to 300. And 26 is closer to 30 than to 20, so we round it to 30.
So 347 × 26 is about 300 × 30 = 9,000. You can estimate before you calculate to know what size the answer should be, or after, to check your answer is sensible. If you round both numbers up, the estimate is bigger than the real answer. If you round both down, it is smaller.
Other ways to check. (1) The last digit. The last digit of the answer comes from multiplying the last digits: 37 × 43 ends in the last digit of 7 × 3 = 21, which is 1. (2) Use a fact you know. If 23 × 45 = 1,035, then 23 × 46 = 1,035 + 23 = 1,058 (one more lot of 23) and 230 × 45 = 10,350. (3) Use the inverse (opposite). If 23 × 45 = 1,035, then 1,035 ÷ 45 = 23.
Key ideas
- Estimate by rounding each number to one digit followed by zeros, then multiply the rounded numbers.
- Use ≈ for “is about equal to”.
- If both numbers are rounded up, the estimate is too big. If both are rounded down, it is too small.
- The last digit of the answer = the last digit of (last digit × last digit). A different last digit means the answer is wrong.
- Use a known fact: one more lot adds the number; one fewer lot takes it away. Multiplying and dividing are inverses.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Round each number to one digit followed by zeros, then multiply the rounded numbers.
- Round each number. Keep the first digit and make the others zeros. Rounded numbers are easy to multiply.38 to the nearest 10 is 40
52 to the nearest 10 is 50 - Multiply the rounded numbers. Use a times table fact and put the zeros back.40 × 50: 4 × 5 = 20, then 2 zeros. 2,000
- Write the estimate. Use the “about equal to” sign.38 × 52 ≈ 2,000
- Think: is it too big or too small? It helps to know which way the rounding went.We rounded one number up and one number down, so we cannot tell for sure whether the real answer is bigger or smaller than the estimate.
Check. The real answer is 1,976, which is close to 2,000. ✓
Watch out. An estimate is not exact, so use ≈ (about equal to) and not =.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Round each number to one digit followed by zeros, then multiply the rounded numbers.
- Round each number. Keep the first digit and make the others zeros. Rounded numbers are easy to multiply.3,872 to the nearest 1,000 is 4,000
- Multiply the rounded numbers. Use a times table fact and put the zeros back.4,000 × 6: 4 × 6 = 24, then 3 zeros. 24,000
- Write the estimate. Use the “about equal to” sign.3,872 × 6 ≈ 24,000
- Think: is it too big or too small? It helps to know which way the rounding went.We rounded up, so the numbers we multiplied are bigger than the real ones. The estimate is a bit too big (an overestimate).
Check. The real answer is 23,232, which is close to 24,000. ✓
Watch out. An estimate is not exact, so use ≈ (about equal to) and not =.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Round each number to one digit followed by zeros, then multiply the rounded numbers.
- Round each number. Keep the first digit and make the others zeros. Rounded numbers are easy to multiply.612 to the nearest 100 is 600
29 to the nearest 10 is 30 - Multiply the rounded numbers. Use a times table fact and put the zeros back.600 × 30: 6 × 3 = 18, then 3 zeros. 18,000
- Write the estimate. Use the “about equal to” sign.612 × 29 ≈ 18,000
- Think: is it too big or too small? It helps to know which way the rounding went.We rounded one number up and one number down, so we cannot tell for sure whether the real answer is bigger or smaller than the estimate.
Check. The real answer is 17,748, which is close to 18,000. ✓
Watch out. An estimate is not exact, so use ≈ (about equal to) and not =.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Estimate by rounding each number, then pick the option that matches.
- Round each number. Keep the first digit and make the others zeros.49 to the nearest 10 is 50
31 to the nearest 10 is 30 - Multiply the rounded numbers. Use a times table fact and put the zeros back.50 × 30: 5 × 3 = 15, then 2 zeros. 1,500
- Find the matching answer. Check every option. Look at the number of zeros carefully.
Option Answer Matches? A 150 no B 1,500 yes C 15,000 no D 2,500 no - Write the answer. B (1,500)
Check. The real answer is 1,519, which is close to 1,500. ✓
Watch out. Options often differ only in the number of zeros. Count them: 300 × 30 has three zeros in the product (9,000).
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Estimate first. If the given answer is far from the estimate, it must be wrong.
- Estimate the answer. Round each number and multiply.48 ≈ 50 and 52 ≈ 50
50 × 50 = 2,500 - Compare with the answer in the question. A sensible answer is close to the estimate.Estimate: 2,500. Given answer: 3,496.
- Decide. 3,496 is much bigger than 2,500, so the answer cannot be right.
Check. The real answer is 2,496. ✓
Watch out. An estimate can show that an answer is WRONG. It cannot prove that an answer is exactly right.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Check the last digit. The ones digit of the answer comes from multiplying the two ones digits.
- Look at the ones digits of the numbers being multiplied. The ones digit of the answer only depends on the ones digits of the two numbers.37 ends in 7. 43 ends in 3.
- Multiply the ones digits. The ones digit of that answer is the ones digit we expect.7 × 3 = 21, so the answer should end in 1
- Look at the last digit of the given answer. 1,592 ends in 2.
- Decide. 2 is not the same as 1. The answer must be wrong.
Check. Work it out: 37 × 43 = 1,591. ✓
Watch out. A matching last digit does not prove the answer is right. A different last digit proves it is wrong.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Use the fact you are given: 23 × 45 = 1,035. Look at how the new calculation is different, and change the answer in the same way.
- Compare the two calculations. The new calculation has one more lot of 45.23 × 45 → 24 × 45
- Work out the difference. One extra lot adds that number to the answer.1,035 + 45 = 1,080
- Write the answer. 24 × 45 = 1,080
Check. Estimate: 24 is about 20; 45 is about 50. 20 × 50 = 1,000. Our answer, 1,080, is close to 1,000. ✓
Watch out. Work out HOW the numbers changed. Adding 1 to a number adds one lot of the other number, not 1.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Use the fact you are given: 23 × 45 = 1,035. Look at how the new calculation is different, and change the answer in the same way.
- Compare the two calculations. One number has been multiplied by 10.23 × 45 → 230 × 45
- Multiply the answer by 10 as well. If one number is 10 times bigger, the answer is 10 times bigger. Move the digits.1,035 × 10 = 10,350
- Write the answer. 230 × 45 = 10,350
Check. Estimate: 230 is about 200; 45 is about 50. 200 × 50 = 10,000. Our answer, 10,350, is close to 10,000. ✓
Watch out. Work out HOW the numbers changed. Adding 1 to a number adds one lot of the other number, not 1.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Use a rounded number that is easy to multiply. If 4,982 is less than 5,000, then 4,982 × 3 is less than 5,000 × 3.
- Choose an easy number to compare with. 5,000 is close to 4,982 and easy to multiply.4,982 is less than 5,000
- Multiply the easy number by 3. This gives a number to compare with.5,000 × 3 = 15,000
- Write your conclusion. If you multiply a smaller number by 3, the answer is smaller too.So 4,982 × 3 is less than 15,000.
Check. The exact answer is 14,946, which is less than 15,000. ✓
Watch out. In a “show that” question you must write a sentence at the end to explain your conclusion.
Watch out
An estimate cannot prove that an answer is exactly right, but a good estimate can prove an answer is wrong. Round to one digit followed by zeros (not just anywhere), and count the zeros carefully: 30 × 400 is 12,000 (three zeros), not 1,200.Exercise 4E Estimating and checking
Non-calculatorKey words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Round each number to one digit followed by zeros, then multiply the rounded numbers.
- Round each number. Keep the first digit and make the others zeros. Rounded numbers are easy to multiply.38 to the nearest 10 is 40
52 to the nearest 10 is 50 - Multiply the rounded numbers. Use a times table fact and put the zeros back.40 × 50: 4 × 5 = 20, then 2 zeros. 2,000
- Write the estimate. Use the “about equal to” sign.38 × 52 ≈ 2,000
- Think: is it too big or too small? It helps to know which way the rounding went.We rounded one number up and one number down, so we cannot tell for sure whether the real answer is bigger or smaller than the estimate.
Check. The real answer is 1,976, which is close to 2,000. ✓
Watch out. An estimate is not exact, so use ≈ (about equal to) and not =.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Round each number to one digit followed by zeros, then multiply the rounded numbers.
- Round each number. Keep the first digit and make the others zeros. Rounded numbers are easy to multiply.71 to the nearest 10 is 70
29 to the nearest 10 is 30 - Multiply the rounded numbers. Use a times table fact and put the zeros back.70 × 30: 7 × 3 = 21, then 2 zeros. 2,100
- Write the estimate. Use the “about equal to” sign.71 × 29 ≈ 2,100
- Think: is it too big or too small? It helps to know which way the rounding went.We rounded one number up and one number down, so we cannot tell for sure whether the real answer is bigger or smaller than the estimate.
Check. The real answer is 2,059, which is close to 2,100. ✓
Watch out. An estimate is not exact, so use ≈ (about equal to) and not =.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Round each number to one digit followed by zeros, then multiply the rounded numbers.
- Round each number. Keep the first digit and make the others zeros. Rounded numbers are easy to multiply.412 to the nearest 100 is 400
18 to the nearest 10 is 20 - Multiply the rounded numbers. Use a times table fact and put the zeros back.400 × 20: 4 × 2 = 8, then 3 zeros. 8,000
- Write the estimate. Use the “about equal to” sign.412 × 18 ≈ 8,000
- Think: is it too big or too small? It helps to know which way the rounding went.We rounded one number up and one number down, so we cannot tell for sure whether the real answer is bigger or smaller than the estimate.
Check. The real answer is 7,416, which is close to 8,000. ✓
Watch out. An estimate is not exact, so use ≈ (about equal to) and not =.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Round each number to one digit followed by zeros, then multiply the rounded numbers.
- Round each number. Keep the first digit and make the others zeros. Rounded numbers are easy to multiply.2,986 to the nearest 1,000 is 3,000
- Multiply the rounded numbers. Use a times table fact and put the zeros back.3,000 × 4: 3 × 4 = 12, then 3 zeros. 12,000
- Write the estimate. Use the “about equal to” sign.2,986 × 4 ≈ 12,000
- Think: is it too big or too small? It helps to know which way the rounding went.We rounded up, so the numbers we multiplied are bigger than the real ones. The estimate is a bit too big (an overestimate).
Check. The real answer is 11,944, which is close to 12,000. ✓
Watch out. An estimate is not exact, so use ≈ (about equal to) and not =.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Estimate by rounding each number, then pick the option that matches.
- Round each number. Keep the first digit and make the others zeros.49 to the nearest 10 is 50
31 to the nearest 10 is 30 - Multiply the rounded numbers. Use a times table fact and put the zeros back.50 × 30: 5 × 3 = 15, then 2 zeros. 1,500
- Find the matching answer. Check every option. Look at the number of zeros carefully.
Option Answer Matches? A 150 no B 1,500 yes C 15,000 no D 2,500 no - Write the answer. B (1,500)
Check. The real answer is 1,519, which is close to 1,500. ✓
Watch out. Options often differ only in the number of zeros. Count them: 300 × 30 has three zeros in the product (9,000).
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Estimate by rounding each number, then pick the option that matches.
- Round each number. Keep the first digit and make the others zeros.612 to the nearest 100 is 600
19 to the nearest 10 is 20 - Multiply the rounded numbers. Use a times table fact and put the zeros back.600 × 20: 6 × 2 = 12, then 3 zeros. 12,000
- Find the matching answer. Check every option. Look at the number of zeros carefully.
Option Answer Matches? A 1,200 no B 12,000 yes C 120,000 no D 18,000 no - Write the answer. B (12,000)
Check. The real answer is 11,628, which is close to 12,000. ✓
Watch out. Options often differ only in the number of zeros. Count them: 300 × 30 has three zeros in the product (9,000).
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Estimate by rounding each number, then pick the option that matches.
- Round each number. Keep the first digit and make the others zeros.3,945 to the nearest 1,000 is 4,000
- Multiply the rounded numbers. Use a times table fact and put the zeros back.4,000 × 6: 4 × 6 = 24, then 3 zeros. 24,000
- Find the matching answer. Check every option. Look at the number of zeros carefully.
Option Answer Matches? A 2,400 no B 24,000 yes C 240 no D 18,000 no - Write the answer. B (24,000)
Check. The real answer is 23,670, which is close to 24,000. ✓
Watch out. Options often differ only in the number of zeros. Count them: 300 × 30 has three zeros in the product (9,000).
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Estimate first. If the given answer is far from the estimate, it must be wrong.
- Estimate the answer. Round each number and multiply.48 ≈ 50 and 52 ≈ 50
50 × 50 = 2,500 - Compare with the answer in the question. A sensible answer is close to the estimate.Estimate: 2,500. Given answer: 3,496.
- Decide. 3,496 is much bigger than 2,500, so the answer cannot be right.
Check. The real answer is 2,496. ✓
Watch out. An estimate can show that an answer is WRONG. It cannot prove that an answer is exactly right.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Estimate first. If the given answer is far from the estimate, it must be wrong.
- Estimate the answer. Round each number and multiply.23 ≈ 20 and 41 ≈ 40
20 × 40 = 800 - Compare with the answer in the question. A sensible answer is close to the estimate.Estimate: 800. Given answer: 943.
- Decide. 943 is close to 800, so the answer could be right.
Check. The real answer is 943. ✓
Watch out. An estimate can show that an answer is WRONG. It cannot prove that an answer is exactly right.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Estimate first. If the given answer is far from the estimate, it must be wrong.
- Estimate the answer. Round each number and multiply.612 ≈ 600 and 19 ≈ 20
600 × 20 = 12,000 - Compare with the answer in the question. A sensible answer is close to the estimate.Estimate: 12,000. Given answer: 11,628.
- Decide. 11,628 is close to 12,000, so the answer could be right.
Check. The real answer is 11,628. ✓
Watch out. An estimate can show that an answer is WRONG. It cannot prove that an answer is exactly right.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Estimate first. If the given answer is far from the estimate, it must be wrong.
- Estimate the answer. Round each number and multiply.304 ≈ 300 and 28 ≈ 30
300 × 30 = 9,000 - Compare with the answer in the question. A sensible answer is close to the estimate.Estimate: 9,000. Given answer: 851.
- Decide. 851 is much smaller than 9,000, so the answer cannot be right.
Check. The real answer is 8,512. ✓
Watch out. An estimate can show that an answer is WRONG. It cannot prove that an answer is exactly right.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Check the last digit. The ones digit of the answer comes from multiplying the two ones digits.
- Look at the ones digits of the numbers being multiplied. The ones digit of the answer only depends on the ones digits of the two numbers.37 ends in 7. 43 ends in 3.
- Multiply the ones digits. The ones digit of that answer is the ones digit we expect.7 × 3 = 21, so the answer should end in 1
- Look at the last digit of the given answer. 1,592 ends in 2.
- Decide. 2 is not the same as 1. The answer must be wrong.
Check. Work it out: 37 × 43 = 1,591. ✓
Watch out. A matching last digit does not prove the answer is right. A different last digit proves it is wrong.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Check the last digit. The ones digit of the answer comes from multiplying the two ones digits.
- Look at the ones digits of the numbers being multiplied. The ones digit of the answer only depends on the ones digits of the two numbers.126 ends in 6. 34 ends in 4.
- Multiply the ones digits. The ones digit of that answer is the ones digit we expect.6 × 4 = 24, so the answer should end in 4
- Look at the last digit of the given answer. 4,282 ends in 2.
- Decide. 2 is not the same as 4. The answer must be wrong.
Check. Work it out: 126 × 34 = 4,284. ✓
Watch out. A matching last digit does not prove the answer is right. A different last digit proves it is wrong.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Check the last digit. The ones digit of the answer comes from multiplying the two ones digits.
- Look at the ones digits of the numbers being multiplied. The ones digit of the answer only depends on the ones digits of the two numbers.58 ends in 8. 67 ends in 7.
- Multiply the ones digits. The ones digit of that answer is the ones digit we expect.8 × 7 = 56, so the answer should end in 6
- Look at the last digit of the given answer. 3,886 ends in 6.
- Decide. 6 is the same as 6. The last digit fits, so the answer might be right (but the other digits could still be wrong).
Check. Work it out: 58 × 67 = 3,886. ✓
Watch out. A matching last digit does not prove the answer is right. A different last digit proves it is wrong.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Check the last digit. The ones digit of the answer comes from multiplying the two ones digits.
- Look at the ones digits of the numbers being multiplied. The ones digit of the answer only depends on the ones digits of the two numbers.215 ends in 5. 48 ends in 8.
- Multiply the ones digits. The ones digit of that answer is the ones digit we expect.5 × 8 = 40, so the answer should end in 0
- Look at the last digit of the given answer. 10,330 ends in 0.
- Decide. 0 is the same as 0. The last digit fits, so the answer might be right (but the other digits could still be wrong).
Check. Work it out: 215 × 48 = 10,320. So 10,330 is wrong, even though its last digit matched! ✓
Watch out. A matching last digit does not prove the answer is right. A different last digit proves it is wrong.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Use the fact you are given: 23 × 45 = 1,035. Look at how the new calculation is different, and change the answer in the same way.
- Compare the two calculations. The new calculation has one more lot of 23.23 × 45 → 23 × 46
- Work out the difference. One extra lot adds that number to the answer.1,035 + 23 = 1,058
- Write the answer. 23 × 46 = 1,058
Check. Estimate: 23 is about 20; 46 is about 50. 20 × 50 = 1,000. Our answer, 1,058, is close to 1,000. ✓
Watch out. Work out HOW the numbers changed. Adding 1 to a number adds one lot of the other number, not 1.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Use the fact you are given: 23 × 45 = 1,035. Look at how the new calculation is different, and change the answer in the same way.
- Compare the two calculations. The new calculation has one more lot of 45.23 × 45 → 24 × 45
- Work out the difference. One extra lot adds that number to the answer.1,035 + 45 = 1,080
- Write the answer. 24 × 45 = 1,080
Check. Estimate: 24 is about 20; 45 is about 50. 20 × 50 = 1,000. Our answer, 1,080, is close to 1,000. ✓
Watch out. Work out HOW the numbers changed. Adding 1 to a number adds one lot of the other number, not 1.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Use the fact you are given: 23 × 45 = 1,035. Look at how the new calculation is different, and change the answer in the same way.
- Compare the two calculations. One number has been multiplied by 10.23 × 45 → 230 × 45
- Multiply the answer by 10 as well. If one number is 10 times bigger, the answer is 10 times bigger. Move the digits.1,035 × 10 = 10,350
- Write the answer. 230 × 45 = 10,350
Check. Estimate: 230 is about 200; 45 is about 50. 200 × 50 = 10,000. Our answer, 10,350, is close to 10,000. ✓
Watch out. Work out HOW the numbers changed. Adding 1 to a number adds one lot of the other number, not 1.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Use the inverse operation: a multiplication fact gives two division facts.
- Spot the connection. Multiplying and dividing undo each other. If 23 × 45 = 1,035, then 1,035 split into groups of 45 gives 23 groups.23 × 45 = 1,035
- Write the division fact. Both facts use the same three numbers.1,035 ÷ 45 = 23 and 1,035 ÷ 23 = 45
- Write the one asked for. 1,035 ÷ 45 = 23
Check. 23 × 45 = 1,035. ✓
Watch out. Division is the opposite of multiplication, so you do not need to work it out again.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Use the fact you are given: 12 × 15 = 180. Look at how the new calculation is different, and change the answer in the same way.
- Compare the two calculations. 30 is double 15.12 × 15 → 12 × 30
- Double the answer. If one number is doubled, the answer is doubled.180 × 2 = 360
- Write the answer. 12 × 30 = 360
Check. Estimate: 12 is about 10. 10 × 30 = 300. Our answer, 360, is close to 300. ✓
Watch out. Work out HOW the numbers changed. Adding 1 to a number adds one lot of the other number, not 1.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Use the fact you are given: 12 × 15 = 180. Look at how the new calculation is different, and change the answer in the same way.
- Compare the two calculations. 6 is half of 12.12 × 15 → 6 × 15
- Halve the answer. If one number is halved, the answer is halved.180 ÷ 2 = 90
- Write the answer. 6 × 15 = 90
Check. Estimate: 6 × 15 is close to the given fact. ✓
Watch out. Work out HOW the numbers changed. Adding 1 to a number adds one lot of the other number, not 1.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Use the fact you are given: 12 × 15 = 180. Look at how the new calculation is different, and change the answer in the same way.
- Compare the two calculations. One number has been multiplied by 10.12 × 15 → 120 × 15
- Multiply the answer by 10 as well. If one number is 10 times bigger, the answer is 10 times bigger. Move the digits.180 × 10 = 1,800
- Write the answer. 120 × 15 = 1,800
Check. Estimate: 120 is about 100; 15 is about 20. 100 × 20 = 2,000. Our answer, 1,800, is close to 2,000. ✓
Watch out. Work out HOW the numbers changed. Adding 1 to a number adds one lot of the other number, not 1.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Use the fact you are given: 12 × 15 = 180. Look at how the new calculation is different, and change the answer in the same way.
- Compare the two calculations. The new calculation has one fewer lot of 15.12 × 15 → 11 × 15
- Work out the difference. One fewer lot takes that number away from the answer.180 − 15 = 165
- Write the answer. 11 × 15 = 165
Check. Estimate: 11 is about 10; 15 is about 20. 10 × 20 = 200. Our answer, 165, is close to 200. ✓
Watch out. Work out HOW the numbers changed. Adding 1 to a number adds one lot of the other number, not 1.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Estimate first by rounding. Then work out the exact answer, and compare it with your estimate.
- Estimate first. Round each number to one digit followed by zeros. This tells us roughly what the answer should be.28 ≈ 30 and 67 ≈ 70
30 × 70 = 2,100 - Split the multiplier. Multiplying by 67 means multiplying by 7 and by 60, then adding. Write the numbers in columns.67 = 60 + 7, so 28 × 67 = (28 × 7) + (28 × 60)
2 8 × 6 7 - Row 1: multiply by the 7 (the ones digit). This is short multiplication, starting with the ones. Work out 28 × 7.Ones: 8 × 7 = 56. Write 6 and carry 5.
Tens: 2 × 7 = 14, and 14 + 5 (carried) = 19. Write 9 and carry 1.
Hundreds: nothing left to multiply. Write the carried 1.
Row 1 = 196 - Row 2: write a 0 first, then multiply by the 6. The 6 is really 60, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 28 by 6.Write 0 in the ones column.
Ones: 8 × 6 = 48. Write 8 and carry 4.
Tens: 2 × 6 = 12, and 12 + 4 (carried) = 16. Write 6 and carry 1.
Hundreds: nothing left to multiply. Write the carried 1.
Row 2 = 168 with a 0 on the end = 1,680 - Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.
2 8 × 6 7 1 9 6 28 × 7 + 1 6 8 0 28 × 60 1 8 7 6
196 + 1,680 = 1,876 - Compare the answer with the estimate. They should be close.1,876 is close to 2,100. ✓
- Write the answer. 1,876
Check. Estimate: 30 × 70 = 2,100, and 1,876 is close. ✓
Watch out. If your answer is nowhere near the estimate, check your working (carries and place holders).
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Estimate first by rounding. Then work out the exact answer, and compare it with your estimate.
- Estimate first. Round each number to one digit followed by zeros. This tells us roughly what the answer should be.312 ≈ 300 and 45 ≈ 50
300 × 50 = 15,000 - Split the multiplier. Multiplying by 45 means multiplying by 5 and by 40, then adding. Write the numbers in columns.45 = 40 + 5, so 312 × 45 = (312 × 5) + (312 × 40)
3 1 2 × 4 5 - Row 1: multiply by the 5 (the ones digit). This is short multiplication, starting with the ones. Work out 312 × 5.Ones: 2 × 5 = 10. Write 0 and carry 1.
Tens: 1 × 5 = 5, and 5 + 1 (carried) = 6. Write 6.
Hundreds: 3 × 5 = 15. Write 5 and carry 1.
Thousands: nothing left to multiply. Write the carried 1.
Row 1 = 1,560 - Row 2: write a 0 first, then multiply by the 4. The 4 is really 40, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 312 by 4.Write 0 in the ones column.
Ones: 2 × 4 = 8. Write 8.
Tens: 1 × 4 = 4. Write 4.
Hundreds: 3 × 4 = 12. Write 2 and carry 1.
Thousands: nothing left to multiply. Write the carried 1.
Row 2 = 1,248 with a 0 on the end = 12,480 - Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.
3 1 2 × 4 5 1 5 6 0 312 × 5 + 1 2 4 8 0 312 × 40 1 4 0 4 0
1,560 + 12,480 = 14,040 - Compare the answer with the estimate. They should be close.14,040 is close to 15,000. ✓
- Write the answer. 14,040
Check. Estimate: 300 × 50 = 15,000, and 14,040 is close. ✓
Watch out. If your answer is nowhere near the estimate, check your working (carries and place holders).
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Round the price UP to a number that is easy to multiply. If the total is still within the budget, then the real total is too.
- Round the price up. 69p is a little less than 70p. Rounding up makes the cost bigger, so if it still fits, the real cost fits.69p → 70p
- Multiply. 28 items at 70p each.28 × 70p = 1,960p
- Compare with the money. £20 is 2,000p.1,960p is less than 2,000p, and the real cost is even less than that.
- Write the answer. Yes, £20 is enough.
Check. The exact cost is 28 × 69p = 1,932p = £19.32, which is less than £20. ✓
Watch out. Round UP when you want to be sure you have enough money.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Round each number to one digit followed by zeros, then multiply. Include the unit.
- Round both numbers. We only need an estimate, so use easy numbers.38 km ≈ 40 km and 52 days ≈ 50 days
- Choose the operation. The same distance every day for 50 days: multiply.40 × 50
- Multiply the rounded numbers. 4 × 5 = 20, then put on the two zeros.40 × 50 = 2,000
- Write the estimate. Include the unit.about 2,000 km
Check. The exact answer is 38 × 52 = 1,976 km, which is close to 2,000 km. ✓
Watch out. Say “about”: an estimate is not exact.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Think about which way each number was rounded. If both numbers were rounded up, the estimate is bigger than the real answer.
- Compare Amir’s numbers with the real numbers. Look at each number: was it rounded up or down?348 → 350 (up) and 47 → 50 (up)
- Decide what that does to the answer. Bigger numbers multiplied together make a bigger product. Smaller numbers make a smaller product.Both numbers were rounded up, so the estimate is bigger than the real answer.
- Check with the real answer. Work it out to see.348 × 47 = 16,356, and the estimate is 17,500, which is more.
Check. 17,500 is greater than 16,356. ✓
Watch out. If one number goes up and the other goes down, the errors partly cancel, so you cannot be sure.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Use a rounded number that is easy to multiply. If 6,018 is more than 6,000, then 6,018 × 5 is more than 6,000 × 5.
- Choose an easy number to compare with. 6,000 is close to 6,018 and easy to multiply.6,018 is more than 6,000
- Multiply the easy number by 5. This gives a number to compare with.6,000 × 5 = 30,000
- Write your conclusion. If you multiply a bigger number by 5, the answer is bigger too.So 6,018 × 5 is more than 30,000.
Check. The exact answer is 30,090, which is more than 30,000. ✓
Watch out. In a “show that” question you must write a sentence at the end to explain your conclusion.
Key words. To estimate is to find an answer that is close, but not exact. To round a number is to change it to a nearby number that is easier to use. A product is the answer to a multiplication.
Method. Work out the correct answer yourself first. Then compare it with the answer in the question to find the mistake.
- Work it out carefully. Do the question again, one step at a time.412 is about 400 (nearest hundred) and 38 is about 40 (nearest ten).
400 × 40 = 16,000 - Compare with the answer in the question. Zainab rounded 38 to 30, then worked out 400 × 30 = 12,000.
- Find the mistake. 38 is closer to 40 than to 30, so it should round to 40. Rounding 38 down to 30 changes the number by 8, which is too much, so her estimate is much too small.
- Write the correct answer. A better estimate is 16,000.
Check. The exact answer is 412 × 38 = 15,656, which is close to 16,000. ✓
Watch out. Round to the NEAREST multiple: 38 is 2 away from 40 but 8 away from 30.
Key words. To compensate is to make up for a change. Here we multiply by an easy number that is a little too big or too small, then correct the answer.
Method. Work out each product. You can use 50 × 51 and a grid, or compare with 50 × 50.
- Work out 50 × 50. This is an easy multiplication.50 × 50 = 2,500
- Work out 49 × 51 by compensating. 49 is one less than 50, so take away one lot of 51 from 50 × 51.50 × 51 = 2,550, then 2,550 − 51 = 2,499
- Compare. Both numbers add to 100, but they are different distances apart.2,500 is more than 2,499, so 50 × 50 is bigger.
Check. Grid: 49 × 51 = (50 − 1) × (50 + 1) = 2,500 + 50 − 50 − 1 = 2,499. ✓
Watch out. The numbers are very close, so an estimate cannot decide it. Work out both exactly.