How to Estimate in Maths

How to Estimate in Maths

Estimation means to simplify numbers in a calculation in order to get a close answer more quickly and easily than doing the full calculation.

• Estimating means to change a number to another number that is close to it.
• Estimating is used to make calculations easier and it gives us an answer that is close to the actual answer.
• 2.8 can be estimated to be close to 3.
• 5.1 can be estimated to be close to 5.
• The sum of 2.8 + 5.1 can be estimated to be 3 + 5.
• 2.8 + 5.1 = 7.9 which is the exact answer.
• This is very close to 3 + 5 = 8, our estimated answer.

Look at the digit to the right of the digit being estimated.

If this digit is 5 or more, round up.

If this digit is 4 or less, round down.

• The second digit of 33 is a 3, which is ‘4 or less’.
• Therefore we round 33 down to 30 when estimating.
• The second digit of 19 is a 9, which is ‘5 or more’.
• Therefore we round 19 up to 20 when estimating.
• 33 – 19 = 14 which is the exact answer.
• 30 – 20 = 10, which is an estimated answer.

Estimating

Why Estimating is Important

In maths, estimation means to simplify numbers in a calculation in order to get a close answer more quickly and easily than doing the full calculation. The benefits of estimation are that it can often be completed mentally and the result can be used to check the result of a calculation.

For example, if you need to buy 5 pens for $2.99 each, it is easier to find 5 Ã—$3 than work out 5 Ã— $2.99. The correct cost is$14.95 which is very close to the estimation of $15. This estimation can be completed quickly and easily in your head. How to Estimate to the Nearest Integer Estimating a number to the nearest integer means to find the nearest whole number to it. Look at the digit in the tenths place. If it is 5 or more, round up. If it is 4 or less round down. For example, 6.27 rounds down to 6 because there is a 2 in the tenths place. An integer is a whole number. When rounding to the nearest integer, only look at the digit immediately after the decimal point in the tenths column to decide whether to round up or down. For example, estimate 2.83 to the nearest integer. The digit in the tenths column is 8. This is ‘5 or more’ and so we round up. We round 2.87 up to 3 which means that we estimate 2.87 to be 3. How to Estimate to the Nearest Ten To estimate a number to the nearest ten, look at the digit in the ones column. If it is 5 or more, round up. If it is 4 or less, round down. For example, 14 rounds down to 10 because there is a 4 in the ones column. For example, estimate 55 to the nearest ten. There is a 5 in the ones column and so, we round up. How to Estimate to the Nearest Hundred To estimate a number to the nearest hundred, look at the digit in the tens column. If it is 5 or more, round up. If it is 4 or less, round down. For example, 247 rounds down to 200 because there is a 4 in the tens column. For example, estimating 1363 to the nearest hundred is 1400 because 6 in the tens column is ‘5 or more’. How to Estimate an Answer When estimating an answer use the following rules: • Focus on the digits at the start of each number as they have a greater impact on the answer. • Round each number greater than one to the nearest whole number, ten, hundred or thousand. • Round any number less than one to the nearest fractional amount. • Round all numbers before performing the calculation. For example, estimate the size of the answer to 39 Ã— 4.85. 39 rounds up to 40. Multiples of 10 are generally easier to multiply. 4.85 rounds up to 5. Five is chosen as it is also an easier number to multiply. Since 4 Ã— 5 = 20, 40 Ã— 5 = 200. We need to add another zero. Here is an example of estimating the total cost of a shopping list. When estimating money, round each amount to the nearest whole number. •$0.90 rounds up to $1. •$1.25 rounds down to $1. •$2.87 rounds up to $3. •$6.10 rounds down to $6. •$3.22 rounds down to $3. Adding up the total we have$1 + $1 +$3 + $6 +$3 = $14. How to Estimate an Addition To estimate an addition, round all numbers to their first digit before adding them. To do this, look at the second digit of each number. If this digit is 5 or more, round up. If it is 4 or less, round down. For example, estimate the addition of 48 + 51. The second digit of 48 is 8, therefore we round it up to 50. The second digit of 51 is 1, therefore we round it down to 50. Perform the addition after the rounding. 50 + 50 = 100, therefore an estimate to 48 + 51 is 100. The correct answer is 99, which is only 1 off 100. Here is another example of estimating an addition. Estimate 384 + 209. The second digit of 384 is 8, therefore it rounds up to 400. The second digit of 209 is 0, therefore it rounds down to 200. 400 + 200 = 600, therefore the estimate to the addition of 384 + 209 = 600. The correct answer to 384 + 209 is 593, which is only 7 away from the estimate of 600. How to Estimate a Subtraction To estimate a subtraction, round each number to its first digit and then subtract. To do this, look at the second digit of each number. If it is 5 or more, round up. If it is 4 or less, round down. For example, estimate the subtraction of 73 – 29. The second digit of 73 is a 3, therefore 73 rounds down to 70. The second digit of 29 is 9, therefore 29 rounds up to 30. The estimate of 73 – 29 is 70 – 30, which equals 40. The correct answer to 73 – 29 is 44, which is 4 away from the estimated answer of 40. Subtraction is used to find a difference. Find the estimated difference between 988 and 674. The second digit of 988 is 8, therefore 988 rounds up to 1000. The second digit of 674 is 7, therefore 674 rounds up to 700. 1000 – 700 = 300, therefore the estimated difference between 988 and 674 is 300. The difference between 988 and 674 is 314. This is 14 away from the estimated difference of 300. How to Estimate Multiplication To estimate the answer to a multiplication, round the numbers to their first digit or to easy to multiply digits before multiplying them. For example, 39 Ã— 4.85 can be estimated as 40 Ã— 5 which equals 200. The correct answer is 189.15. Here is another example of estimating multiplication. Estimate 482 Ã— 734. 482 can be estimated as 500. 734 can be estimated as 700. To calculate 500 Ã— 700, multiply 5 Ã— 7 to get 35 and add on the four zeros found in 500 and 700. 500 Ã— 700 = 350000 and so, the estimate to 482 Ã— 734 is 350000. The correct answer is 353788. How to Estimate Division To estimate division, find similar numbers that can be divided exactly. For example 15 Ã· 4 can be estimated as 16 Ã· 4 = 4. 15 is close to 16 and 16 is chosen because it can be divided exactly by 4. For example, estimate the division 194592 Ã· 4126. In this example, 194592 rounds to 200000 and 4126 can be rounded to 4000. We can cancel the three zeros in 4000 with the three zeros in 200000 to leave 200 Ã· 4. 20 Ã· 4 = 5 and so 200 Ã· 4 = 50. Here is another example of estimating division. Estimate 19 Ã· 3. Instead of rounding 19 up to 20, it is best to round it down to 18. We choose the nearest number that can be divided exactly by 3. 18 Ã· 3 = 6 and so, 19 Ã· 3 is just a little larger than 6. 19 Ã· 3 = 6.33. How to Estimate Decimals When calculating with decimals, try to round the decimal number to the nearest whole number. For decimals less than one whole, round the decimal to a number that is equivalent to a simple fraction. Here are some useful decimals and their fraction equivalents. Decimal Equivalent Fraction 0.1 1/10 0.2 1/5 0.25 1/4 0.33 1/3 0.4 2/5 0.5 1/2 0.6 3/5 0.66 2/3 0.75 3/4 0.8 4/5 For example, estimate 0.26 Ã— 23.87. 0.26 can be estimated as 0.25, which is the same as 1/4. 28.87 can be estimated as 24. 0.26 Ã— 23.87 can be estimated as 1/4 of 24 which equals 6. How to Estimate Decimals When Adding To estimate addition involving decimals, round each decimal to the nearest whole number and then add them. For example, estimate 1.85 + 14.03 + 3.92. 2 + 14 + 4 = 20. How to Estimate Decimal Subtraction To estimate subtraction involving decimals, round each decimal to the nearest whole number and then subtract them. For example, estimate 14.99 – 2.85. Rounding to the nearest whole number this can be estimated as 15 – 3 = 12. How to Estimate Decimal Multiplication To estimate a decimal multiplication: • Round any number greater than one to the nearest whole number. • Round any decimal less than one to a decimal that has a fraction equivalent. For example estimate 0.32 Ã— 59.3. 0.32 can be estimated as 0.33, which is equivalent to 1/3. 59.3 can be estimated as 60. 0.32 Ã— 59.3 can be estimated to be 1/3 of 60, which equals 20. The exact answer is 18.976. How to Estimate Decimal Division To estimate a decimal division, round the numbers so that the division can be done exactly. For example, estimate 7.9 Ã· 2.03. The numbers can be estimated as 8 Ã· 2 and so the division can be estimated to be equal to 4 The exact answer is 3.89. How to Estimate to the Nearest Tenth To estimate a number to the nearest tenth, look at the digit in the hundredths column. If it is 5 or more, add 1 to the digit in the tenths column. If it is 4 or less, keep the tenths digit the same. Remove all digits that follow. For example estimate 0.5814 + 2.632. There is an 8 in the hundredths column of 0.5814 and so, 0.5814 rounds up to 0.6. There is a 3 in the hundredths column of 2.632 and so, 0.2632 rounds down to 2.6. 0.6 + 2.6 = 3.2 and so the estimate for this calculation is 3.2. The correct answer is 3.2134. How to Estimate Percentages To estimate a percentage: • Estimate what 50% is by dividing the total by 2. • Estimate what 10% is by dividing the total by 10. • Estimate what 5% is by finding half of 10%. • Estimate what 1% is by dividing the total by 100. • Use combinations of these percentages to find a similar percentage to that required. For example, estimate 52% of 40. 52% is very similar to 50%. Therefore an estimate of 52% of 40 will be approximately half of 40, which is 20. We know that 52% will be a little larger than 50% and so, we can estimate it as larger than 20. 1% is found by dividing 40 by 100 to get 0.4 and so, doubling this 2% is 0.8. Therefore 52% is 20.8. For example, estimate 16% of$20.

16% is similar to 15% and so, can be estimated by adding 10% and 5%.

10% is found by dividing $20 by 10 to get$2 and then 5% is half of 10%, which is $1. 15% is therefore$2 + $1 =$3. Therefore 16% is a little larger than $3. To find 16%, add 1% to the 15% found previously. 1% is found by dividing$20 by 100 to get $0.20. Therefore 16% is$3.20.

How to Estimate with Fractions

To estimate with fractions, round each fraction to the nearest whole number. If the fraction is less than one whole, compare the numerator to the denominator so that the fraction can be compared to a fraction that is easier to work with.

For example, estimate 14/29 of 14.

14 is exactly half of 28 and so, 14/29 can be estimated as 1/2.

1/2 of 14 is 7.

How to Estimate Fractions When Adding

When estimating adding fractions, compare the numerator to the denominator. If the numerator is close to the denominator, estimate the fraction as one whole. If the numerator is much less than the denominator, estimate the fraction as zero. If the numerator is similar to half of the denominator, estimate the fraction as one half.

For example, estimate 5/6 + 17/20 + 5/9 + 1/11.

5/6 can be estimated as 1 whole because 5 is close to 6.

17/20 can also be estimated as 1 whole because 17 is close to 20.

5/9 can be estimated as one half because 5 is half of 10, which is close to half of 9.

1/11 can be estimated as zero since 1 is much less than 11.

Therefore 5/6 + 17/20 + 5/9 + 1/11 can be estimated as 1 + 1 + 0.5 + 0, which equals 2.5

The correct answer is 2.33, which is close to the estimate of 2.5.

How to Estimate Fractions When Subtracting

To estimate fractions when subtracting, compare the numerator to the denominator. If the numerator is close to the denominator, estimate the fraction as one whole. If the numerator is much less than the denominator, estimate the fraction as zero. If the numerator is similar to half of the denominator, estimate the fraction as one half.

For example, estimate 37/86/11.

7/8 can be estimated as one whole since 7 is close to 8. Therefore 37/8 can be estimated as 4.

6/11 can be estimated as 1/2 since 6 is close to half of 11.

Therefore 37/86/11 can be estimated as 4 – 0.5 which equals 3.5.

The correct answer is approximately 3.33 which is close to the estimate of 3.5.

How to Estimate Fractions When Multiplying

To estimate multiplication with fractions, compare each fraction to one half. If the fraction is greater than or equal to one half, round it up to the next whole number. If the fraction is less than one half, round it down to the previous whole number. Then multiply the whole numbers together.

For example, estimate 44/5 Ã— 21/4.

44/5 can be estimated as 5 since 4/5 is larger than 1/2.

21/4 can be estimated as 2 since 1/4 is less than 1/2.

44/5 Ã— 21/4 can be estimated as 5 Ã— 2 = 10.

The exact answer is 10.8 which is close to the estimate of 10.

How to Estimate Fractions When Dividing

To estimate division with fractions, compare each fraction to one half. If the fraction is greater than or equal to one half, round it up to the next whole number. If the fraction is less than one half, round it down to the previous whole number. Then divide the whole numbers.

For example, estimate 33/4 Ã· 21/5.

33/4 can be estimated as 4 since 3/4 is larger than one half.

21/5 can be estimated as 2 since 1/5 is less than one half.

33/4 Ã· 21/5 can be estimated as 4 Ã· 2 which equals 2.

The correct answer is approximately 1.70 which is close to the estimate of 2.

Now try our lesson on Halving Odd Numbers where we learn how to halve an odd number.

How to Write Numbers in Ascending and Descending Order

How to Write Numbers in Ascending and Descending Order

Ascending order means to write numbers from smallest to largest.

Descending order means to write numbers from largest to smallest.

• Ascending means to go up.
• Numbers written in ascending order increase from smallest to largest.
• Descending means to go down.
• Numbers written in descending order decrease from largest to smallest.
• The numbers 1, 2, 3, 4, 5 are written in ascending order because they get larger.
• The numbers 5, 4, 3, 2, 1 are written in descending order because they get smaller.

• The numbers 3, 1, 8, 2, 0 written in ascending order are 0, 1, 2, 3, 8.
• Zero is smaller than all positive numbers.
• The symbol ‘<' means 'less than'.
• We can write numbers in ascending order like so: 0 < 1 < 2 < 3 < 8.

Supporting Lessons

Ascending and Descending Order

What is Ascending Order

Numbers written in ascending order are arranged from smallest to largest. For example, the numbers 3, 4, 1, 5, 2 written in ascending order are 1, 2, 3, 4, 5. The numbers increase in size.

Ascending means to go up or increase.

Numbers can be written in ascending order using the less than symbol ‘<'. The smaller number is to the left of the sign and the larger number is to the right of the sign. For example, the numbers 1<2<3<4<5<6<7<8<9 are written in ascending order.

Examples of numbers written in ascending order:

• 0 < 1 < 2 < 3 < 4 < 5
• 10 < 11 < 12 < 13 < 14 < 15
• 10 < 20 < 30 < 40 < 50 < 60 < 70 < 80 < 90 < 100
• -10 < -9 < -8 < -7 < -6 < -5 < -4 < -3 < -2 < -1 < 0 < 1
• 0 < 1/10 < 1/5 < 1/4 < 1/3 < 1/2 < 3/4

Writing numbers in ascending order is useful because it allows for the organisation of data. When numbers are written in ascending order, similar values can also be grouped together and extreme values can be found. Putting numbers in ascending order is also necessary for many calculations such as finding the median or quartiles.

What is Descending Order

Numbers written in descending order are arranged from largest to smallest. For example, the numbers 3, 4, 1, 5, 2 written in descending order are 5, 4, 3, 2, 1. The numbers decrease in size.

Descending means to go down. Numbers written in descending order go down.

The ‘greater than’ sign ‘>’ is used to arrange numbers in descending order. For example the numbers 9>8>7>6>5>4>3>2>1 are written in descending order.

Examples of numbers written in descending order:

• 5 > 4 > 3 > 2 > 1 > 0
• 15 > 14 > 13 > 12 > 11 > 10
• 100 > 90 > 80 > 70 > 60 > 50 > 40 > 30 > 20 > 10
• 1 > 0 > -1 > -2 > -3 > -4 > -5
• 3/4 > 1/2 > 1/3 > 1/4 > 0

How to Write Fractions in Ascending Order

• For fractions with the same denominator, the larger the numerator, the larger the fraction.
• For fractions with the same numerator, the smaller the denominator, the larger the fraction.
• For all other fractions, find a common denominator and the larger the numerator, the larger the fraction.

For example, the fractions 1/5, 2/5, 3/5, 4/5, 5/5 are in ascending order.

The fractions have the same denominator and so, they are ordered using the value of their numerator. The larger the numerator, the larger the fraction.

For example, the fractions 1/6, 1/5, 1/4, 1/3, 1/2 are in ascending order.

The fractions have the same numerator and so, they are ordered using their denominator. The smaller the denominator, the larger the fraction.

For example, the fractions 1/6, 2/3, 1/2, 5/12 and 3/4 can be written in ascending order by finding equivalent fractions with the same denominator.

We can write the fractions as 2/12, 5/12, 6/12, 8/12 and 9/12.

Now the fractions have the same denominator, they can be ordered by arranging them according to the size of their numerator.

The fractions in ascending order are 1/6, 5/12, 1/2, 2/3 and 3/4.

How to Write Fractions in Descending Order

Writing fractions in descending order means to put them in order from largest to smallest. To arrange fractions in descending order, find equivalent fractions with common denominators and then order them by their numerators. If fractions have the same denominator, those with the larger numerator are larger fractions.

For example, write the fractions 11/20, 7/10, 4/5, 3/4 and 1/2 in descending order.

These fractions can be written to have a common denominator of 20.

The fractions in descending order are 4/5, 3/4, 7/10, 11/20 and 1/2.

How to Write Decimals in Ascending Order

To arrange decimals in ascending order:

1. Compare each digit in each decimal from left to right.
2. The first decimal to have a larger digit is the larger decimal.
3. If the digits are the same, look at the next digit to the right.

For example, compare the size of the decimals 0.2 and 0.14.

Starting from the left, both decimals have a 0 in the ones column.

So we look at the next digit in the tenths column. 2 is larger than 1 and so 0.2 is larger than 0.14. Therefore 0.14 < 0.2.

In ascending order, we write 0.14, 0.2.

For example, compare 0.3215 and 0.3211.

Starting with the leftmost digits, both decimals have a 0, then a 3, then a 2, then a 1. In the final column, 0.3215 has a 5, whilst 0.3211 has a 1.

Therefore 0.3215 is larger than 0.3211. We can write 0.3211 < 0.3215.

In ascending order we have 0.3211, 0.3215.

To write lists of decimals in ascending order, compare each decimal number to each of the other decimals and write them from smallest to largest.

For example, write the decimals 0.2, 0.12, 0.034 and 0.105.

Each decimal has a ones digit of 0, so we compare the next digit to the right.

0.2 has a 2 in the tenths column, which is larger than 1 or 0 and so, 0.2 is the largest decimal. 0.034 has a 0 in the tenths column and therefore it is the smallest decimal.

Comparing 0.12 and 0.105, 0.12 has a 2 in the next column, whereas 0.105 has a 0. Therefore 0.12 is larger than 0.105.

In ascending order, the decimals are 0.034, 0.105, 0.12 and 0.2.

How to Write Decimals in Descending Order

Writing decimal in descending order means to write them from largest to smallest. To compare the size of decimals, compare each digit from left to right. The decimal with the largest digit is largest. If the digits are equal, compare the next digit to the right.

For example 0.09, 0.7, 1.02 and 0.11 written in descending order is 1.02, 0.7, 0.11 and 0.09.

How to Write Negative Numbers in Ascending Order

To write negative numbers in ascending order, write them from smallest to largest. The bigger the number after the negative sign, the smaller it is. For example, 5 is bigger than 3 but -5 is smaller than -3. An example of negative numbers written in ascending order is -10, -8, -5, -3, -2, -1.

Negative numbers written in ascending order appear backwards compared to positive numbers arranged in ascending order.

The larger the number after the negative sign, the smaller the number.

Negative numbers are always smaller than positive numbers.

For example, the numbers -4, 5, 0, -2 and 1 arranged in ascending order are -4, -2, 0, 1, 5.

To write negative numbers in descending order, write them from largest to smallest. The larger the number after the negative sign, the smaller the number. For example, the numbers -1, -2 ,-3, -4, -5 are written in descending order.

Dates in Ascending Order

To write dates in ascending order, write the oldest dates first and the most recent dates last. For example in ascending order, the date of 01/01/1900 comes before 01/01/2020.

Dates in Descending Order

To write dates in descending order, write the most recent dates first and the oldest dates last. For example in descending order, the date of 01/01/2020 comes before 01/01/1900.

Ascending and Descending Alphabetical Order

Ascending alphabetic order sorts words by their first letter, with ‘A’ first and ‘Z’ last. Descending alphabetic order sorts the words with ‘Z’ first and ‘A’ last. Where words have the same first letter, the next letter is compared to order them.

• Ascending alphabetical order is from A to Z.
• Descending alphabetical order is from Z to A.

For example, the following words are arranged in ascending alphabetical order.

Words in ascending alphabetical order: Alligator, Ant, Anteater, Bear, Cat, Coyote, Deer, Dog.

The following words are arranged in descending alphabetical order.

Words in descending alphabetical order: Zoo, Yoyo, Xylophone, Monkey, Lemon, Kangaroo, Dingo, Aardvark.

Now try our lesson on How to Compare Unlike Fractions where we learn how to compare fractions that have a different denominator.

How to Convert Between Minutes and Hours

How to Convert Between Minutes and Hours

There are 60 minutes in 1 hour.

To convert minutes to hours, divide by 60.

To convert hours to minutes, multiply by 60.

• To convert hours to minutes, multiply by 60.
• 1 hour = 60 minutes.
• 2 hours = 120 minutes.
• 3 hours = 180 minutes.
• 4 hours = 240 minutes.
• 1/2 of an hour = 30 minutes.
• 1/4 of an hour = 15 minutes.

• 2 Ã— 60 = 120, therefore there are 120 minutes in 2 hours.
• 1/2 of 60 is 30, therefore there are 30 minutes in half an hour.
• Therefore the number of minutes in 2 and a half hours is 120 + 30 = 150 minutes.

• To convert minutes to hours, we can divide by 60
• Alternatively, we can subtract lots of 60.
• Each lot of 60 minutes that can be subtracted equals one hour.
• 65 – 60 = 1 remainder 5.
• Therefore 65 minutes = 1 hour 5 minutes.

Supporting Lessons

Converting Units of Time

How to Convert Hours to Minutes

To convert hours to minutes, multiply the number of hours by 60. For example, 3 hours = 180 minutes since 3 Ã— 60 = 180.

The formula to convert hours to minutes is: minutes = hours Ã— 60.

For example, convert 4.5 hours to minutes.

4.5 Ã— 60 = 270, therefore there are 270 minutes in 4.5 hours.

The table below shows some common conversions of hours to minutes.

Hours Minutes
1/60 of an hour 1 minute
1/30 of an hour 2 minutes
1/20 of an hour 3 minutes
1/10 of an hour 6 minutes
1/5 of an hour 12 minutes
1/4 of an hour 15 minutes
1/3 of an hour 20 minutes
1/2 of an hour 30 minutes
1 hour 60 minutes
1 1/2 hours 90 minutes
2 hours 120 minutes
2 1/2 hours 150 minutes
3 hours 180 minutes
3 1/2 hours 210 minutes
4 hours 240 minutes
4 1/2 hours 270 minutes
5 hours 300 minutes

We can combine values from this table to find the number of minutes in a given number of hours.

For example, convert two and a half hours to minutes.

There are 120 minutes in 2 hours.

There are 30 minutes in half an hour.

Therefore there are 150 minutes in 2 and a half hours.

For example, find the number of minutes in 3 hours 40 minutes.

There are 180 minutes in 3 hours.

180 + 40 = 220 minutes and so, there are 220 minutes in 3 hours 40 minutes.

How to Convert Minutes to Hours

To convert minutes to hours, divide the number of minutes by 60. For example, 120 minutes = 2 hours because 120 Ã· 60 = 2.

The formula to convert minutes to hours is: Hours = Number of minutes Ã· 60.

For example, convert 240 minutes to hours.

240 Ã· 60 = 4 and so, there are 4 hours in 240 minutes.

The following table shows the conversion of minutes to hours.

Number of Minutes Hours
15 minutes 0.25 (one quarter) of an hour
20 minutes 0.33 (one third) of an hour
30 minutes 0.5 (one half) of an hour
45 minutes 0.75 (3 quarters) of an hour
60 minutes 1 hour
90 minutes 1.5 (one and a half) hours
120 minutes 2 hours
150 minutes 2.5 (2 and a half) hours
180 minutes 3 hours
210 minutes 3.5 (3 and a half) hours
240 minutes 4 hours
270 minutes 4.5 (4 and a half) hours
300 minutes 5 hours

How to Convert a Time to Hours and Minutes

To convert a time in minutes to hours and minutes:

1. Divide the number of minutes by 60.
2. The number of hours is the integer part of the result.
3. The number of minutes is found by multiplying the decimal part of the number by 60.

For example, convert 150 minutes to hours and minutes.

Step 1. Divide the number of minutes by 60

150 Ã· 60 = 2.5.

This number is made up of 2 + 0.5.

Step 2. The number of hours is the integer part of the result

The integer part of the result is the part of the number in front of the decimal point.

Before the decimal point is a 2. Therefore the number of hours is 2.

Step 3. The number of minutes is found by multiplying the decimal part of the number by 60

The decimal part of the number is 0.5.

0.5 Ã— 60 = 30 and so there are 30 minutes.

150 minutes = 2 hours and 30 minutes.

For example, convert 340 minutes to hours and minutes.

Step 1. Divide the number of minutes by 60

340 Ã· 60 = 5.66.

This is made up of 5 + 0.66.

Step 2. The number of hours is the integer part of the result

The integer part of the result is 5. There are 5 hours.

Step 3. The number of minutes is found by multiplying the decimal part of the number by 60

The decimal part of 5.66 is 0.66.

0.66 Ã— 60 = 40 and so, there are 40 minutes.

340 minutes = 5 hours and 40 minutes.

Alternative Method for Converting a Time to Hours and Minutes

To convert a time in minutes to hours and minutes, subtract multiples of 60 from the number. The number of times 60 can be subtracted is the number of hours and the remainder is the number of minutes.

For example, convert 175 minutes into hours and minutes.

175 – 60 = 115.

115 – 60 = 55.

Therefore we can subtract two lots of 60, which is 2 hours.

The remainder is 55 minutes.

175 minutes = 2 hours 55 minutes.

For example, convert 225 minutes into hours and minutes.

Subtract one hour: 225 – 60 = 165.

Subtract a second hour: 165 – 60 = 105.

Subtract a third hour: 105 – 60 = 45.

225 minutes = 3 hours and 45 minutes.

Minutes to Hours Conversion Chart

Minutes Hours Hours and Minutes
1 min 0.0166 hr 0 hr 1 min
2 min 0.033 hr 0 hr 2 min
3 min 0.05 hr 0 hr 3 min
4 min 0.066 hr 0 hr 4 min
5 min 0.0833 hr 0 hr 5 min
6 min 0.1 hr 0 hr 6 min
7 min 0.1166 hr 0 hr 7 min
8 min 0.133 hr 0 hr 8 min
9 min 0.15 hr 0 hr 9 min
10 min 0.166 hr 0 hr 10 min
15 min 0.25 hr 0 hr 15 min
20 min 0.33 hr 0 hr 20 min
30 min 0.5 hr 0 hr 30 min
40 min 0.66 hr 0 hr 45 min
45 min 0.75 hr 0 hr 45 min
50 min 0.833 hr 0 hr 50 min
60 min 1 hr 1 hr 0 min
70 min 1.166 hr 1 hr 10 min
80 min 1.33 hr 1 hr 20 min
90 min 1.5 hr 1 hr 30 min
100 min 1.66 hr 1 hr 40 min
120 min 2 hr 2 hr 20 min
180 min 3 hr 3 hr 0 min
200 min 3.33 hr 3 hr 20 min
240 min 4 hr 4 hr 0 min
300 min 5 hr 5 hr 0 min
400 min 6.66 hr 6 hr 40 min
500 min 8.33 hr 8 hr 20 min
600 min 10 hr 10 hr 0 min
700 min 11.66 hr 11 hr 40 min
800 min 13.33 hr 13 hr 20 min
900 min 15 hr 15 hr 0 min
1000 min 16.66 hr 16 hr 40 min
1440 min 24 hr 24 hr 0 min
2880 min 48 hr 48 hr 0 min
3000 min 50 hr 50 hr 0 min
6000 min 100 hr 100 hr 0 min
525600 min 8760 hr 8760 hr 0 min

There are 525,600 minutes in one year.

There are 8760 hours in one year.

Now try our lesson on Reading Timetables where we learn how to read a variety of different timetables.

The leading digit of a number is the first digit that is not a zero.

Leading digit approximation only uses this digit and the other digits are replaced with a zero.

• The leading digit is the first non-zero digit in a number.
• The leading digit in 23 is 2 and in 31 is 3.
• Leading digit approximation means to round the number to the leading digit and set the other digits to zero.
• 23 is approximated as 20 and 31 is approximated as 30.
• 23 + 31 = 54 which is the exact answer.
• 20 + 30 = 50 which is the leading digit approximation.

Look at the following digit to decide what the leading digit will round to.

If it is 5 or more, round up. If it is 4 or less, round down.

• The leading digit of 49 is 4.
• We look at the digit after the 4 to decide if we round up or down.
• We look at the 9.
• If it is 5 or more, round up.
• If it is 4 or less, round down.
• 9 is ‘5 or more’ and so we round up.
• We round 49 up to 50.

• 320 has a leading digit of 3
• The next digit is 2, which is ‘4 or less’ so we round 320 down to 300.
• 184 has a leading digit of 1.
• The next digit is 8, which is ‘5 or more’ so we round 184 up to 200.
• The exact answer to 320 – 184 is 136.
• The leading digit approximation to 320 – 184 is 300 – 200 = 100.

Supporting Lessons

What is a Leading Digit of a Number?

The leading digit of a number is the first non-zero digit. For most numbers, the leading digit is simply the first digit. For example, the leading digit of 473 is 4. However, for decimal numbers starting with 0, the leading digit is the first digit that is not zero. For example, the leading digit of 0.0352 is the 3.

The leading digit of 473 is 4.

For example, the leading digit of 23 is 2.

The Leading Digit of a Decimal

The leading digit of a decimal number is the first digit in the number that is not a zero.

The leading digit of 0.0352 is 3.

For example, the leading digit of 0.000104 is 1.

For example, the leading digit of 34.002 is 3. It is simply the first digit because the first digit is not a zero.

How to do Leading Digit Approximation

To round a number to its leading digit, first find the leading digit which is the first non-zero digit. Then look at the digit that comes immediately after it. If it is 5 or more, round the leading digit up to the next number. If it is 4 or less, leave the leading digit alone. Change all other digits to 0.

For example, find the leading digit approximation for 57. The choice is to round down to 50 or round up to 60.

The leading digit is the 5, so we look at the 7 to decide what to do.

7 is ‘5 or more’ so we round up to 60.

For example, use leading digit estimation to round 148. The choice is round down to 100 or up to 200.

The leading digit is 1, so we look at the 4 to decide what to do.

4 is ‘4 or less’ so we round down. The leading digit stays as a 1 and the other digits become 0.

Here is an example of using leading digit estimation on a decimal number 0.0797.

The leading digit is the 7 so we look at the 9 to decide whether to round up or down.

9 is ‘5 or more’ so we round up. The 7 becomes an 8 and we can set all other digits to 0.

0.0797 rounds up to 0.0800.

With decimal numbers, there is no need to write the 0 digits at the end of the number, so the number is written as 0.08.

Leading digit approximation is used to estimate the answers to addition in order to make the addition easier. For example, 23 + 31 can be approximated as 20 + 30 which equals 50. The exact answer is 54.

The addition is made easier using leading digit estimation because only the first digits need to be added rather than adding every digit in the number. The more digits an addition has, the easier it is to use leading digit approximation rather than adding the number fully.

Leading digit estimation can also be used to estimate a subtraction. For example 320 – 184 can be estimated as 300 – 200 which equals 100. The exact answer is 136.

Now try our lesson on Rounding Decimals to the Nearest Whole Number where we learn how to round to the nearest whole number.

How to Read and Understand Stem and Leaf Plots

How to Read and Understand Stem and Leaf Plots

Each number is separated into a stem and a leaf.

Write the last digit of each number in the leaf and the other digits are written in the stem.

• A stem and leaf plot is a table in which numbers are arranged into a ‘stem’ and a ‘leaf’.
• Write the last digit of each number in the leaf.
• All digits in front of the last digit are written in the ‘stem’.
• The numbers must be written from smallest to largest.
• Numbers with the same front digits share a ‘stem’.
• For example, 10| 1 1 2 3 represents the numbers 101, 101, 102 and 103.
• 11| 4 5 represents the numbers 114 and 115.
• 12| 0 2 5 represents the numbers 120, 122 and 125.
• 13| 1 represents the number 131.

• The numbers in a stem and leaf plot are always written from smallest to largest.
• The minimum is the first number in the plot, which is 101.
• The maximum is the last number in the plot, which is 131.
• The modal value is the number that comes up the most, which is 101.

Each number is represented by just one digit, written in the leaf part of the plot.

The numbers are listed from smallest to largest.

• In a stem and leaf plot only the final digit is written in the leaf.
• In this example, the digit after the decimal point is written in the leaf and the digit before the decimal point is the stem.
• 3| 9 represents 3.9.
• The stem of 4| has no numbers in its leaf. This means that there are no numbers that start with a whole number of 4.
• 5| 0 4 7 9 represents the numbers 5.0, 5.4, 4.7 and 5.9.
• 6| 0 2 4 5 8 represents the numbers 6.0, 6.2, 6.4, 6.5 and 6.8.
• 7| 3 5 represents the numbers 7.3 and 7.5.

Supporting Lessons

Stem and Leaf Plots

What is a Stem and Leaf Plot?

A stem and leaf plot is a table in which numbers are represented by one digit, listed in order. Each number is written with its first digits in the ‘stem’ and its final digit listed in the ‘leaf’. A stem and leaf plot is used to organise data visually so that the distribution, skew, outliers and mode can easily be seen.

The numbers in a stem and leaf diagram must be listed in order.

A stem and leaf plot is read by combining the digits in the stem with the digit in the leaf.

Stem Leaf Numbers represented
10 1 1 2 3 101, 101, 102, 103
11 4 5 114, 115
12 0 2 5 120, 122, 125
13 1 131

Stem and leaf plots are used to display numerical data written as a list.

A stem and leaf plot displays all data values and this allows the mode to be easily read.

When making a stem and leaf diagram, a key must always be included. A key is made by writing out one example from the diagram, including units. Keys are particularly useful if the numbers being represented contain many digits, units or are made from decimal numbers.

How to Make a Stem and Leaf Plot

To make a stem and leaf plot:

1. Identify the minimum and maximum values in the data.
2. Create the stem by listing the first digits of the numbers.
3. Draw a line between the stem and the leaves.
4. Represent the numbers from smallest to largest by writing the final digit of each number in the correct leaf.
5. Create a key by writing an example to explain the diagram.

For example, make a stem and leaf plot to represent the following data showing the ages of people who at a park.

15, 1, 28, 7, 5, 33, 41, 4, 10, 35, 24, 2, 32, 6, 11, 4, 12, 5, 37, 30.

Step 1. Identify the minimum and maximum values in the data.

The minimum value is 1 and the maximum value is 41.

Step 2. Create the stem by listing the first digits of the numbers.

The stem is made from all of the digits in the numbers except for the final digit.

In this example, the stem will be the tens place value column.

In the minimum value of 1, there are 0 tens.

In the maximum value of 41, there are 4 tens.

Therefore the stem is a list of numbers from 0 to 4.

The stem is made of 0, 1, 2, 3 and 4.

Step 3. Draw a line between the stem and the leaves.

Step 4. Represent the numbers from smallest to largest by writing the final digit of each number in the correct leaf.

We start with the numbers with zero tens. We list them from smallest to largest.

The first number with zero tens is 1. We place it in the leaf alongside the 0 in the stem.

Now we list the other numbers with zero tens.

The numbers are 2, 4, 4, 5, 5, 6 and 7. They are listed in order in line with the 0 in the stem.

If any numbers are repeated, they are still listed multiple times.

Now we list the numbers with a 1 in the tens column.

We have 10, 11, 12 and 15.

We list the final digits of these numbers in line with the 1 in the stem.

10, 11, 12 and 15 are listed as 1| 0 1 2 5.

We only use the 1 in the tens column once in the stem of the diagram. The numbers are listed by their final digit only.

Now we list the numbers with a 2 in the tens column.

24 and 28 are listed as 2| 4 8.

Now we list the numbers with a 3 in the tens column.

30, 32, 33, 35 and 37 are represented as 3| 0 2 3 5 7.

Now we list the numbers with a 4 in the tens column.

We just have 41 which is listed as 4|1.

Step 5. Create a key by writing an example to explain the diagram.

To make a key on a stem and leaf plot, write any number from the diagram and explain how it is read. For example 1|5 = 15.

Interpreting a Stem and Leaf Plot

On a stem and leaf plot the stem is to the left of the vertical line and the leaf is to the right. Combine the digits in the stem with each digit in the leaf to read each number.

How to Find the Range on a Stem and Leaf Plot

The range is the maximum value subtract the minimum value. On a stem and leaf plot, the minimum is the first value and the maximum is the last value. For example if the maximum value is 41 and the minimum value is 1, the range is 41 – 1 which equals 40.

Here is another example of finding the range on a stem and leaf diagram.

The range= maximum – minimum.

The maximum is $55 and the minimum is$12.

The range = 55 – 12 = 43.

How to Find Quartiles on a Stem and Leaf Plot

• The lower quartile is found at the 1/4 (n+1)th term on a stem and leaf plot.
• The upper quartile is found at the 3/4 (n+1)th term.
• The interquartile range is equal to the upper quartile subtract the lower quartile.

Here is an example of a stem and leaf plot with 19 data values. n = 19.

The lower quartile is found at the 1/4 (n+1)th term. We substitute n = 19 into the formula.

1/4 (19+1) = 5 and so, the lower quartile is found at the 5th term.

The 5th term is 11. The lower quartile is 11.

The upper quartile is found at the 3/4 (n+1)th term. We substitute n = 19 into the formula.

3/4 (19+1) = 15 and so, the lower quartile is found at the 15th term.

The 15th term is 30. The upper quartile is 30.

The interquartile range is equal to the upper quartile – lower quartile.

IQR = 30 – 11.

IQR = 19.

Skewness of a Stem and Leaf Diagram

On a stem and leaf plot, data is skewed when the majority of the data is clustered at the start or end of the plot. If the majority of the data is found at the start of the plot, the data has right (or positive) skew. If the majority of the data is found at the end of the plot, the data has left (or negative skew). If amount of data is the same at each end, it is symmetrical.

To find the skew of a stem and leaf plot, draw a line over the top of the data. Look at the shape of the line drawn. If it is clustered at the start of the data, it is right (or positively) skewed and if it is clustered at the end of the data, it is left (or negatively) skewed.

For example, what is the skew of the following stem and lead plot?

We draw a line over the data to see its distribution.

There is a peak in the data at the start of the plot. Therefore the data is right (or positively) skewed.

Not all stem and leaf plots show skew. Below is an example of a stem and leaf diagram with no skew.

Back-to-Back Stem and Leaf Plots

A back-to-back stem and leaf plot is used to compare two sets of data. One set of data is shown in the leaves to the right of the stem, whilst the other set of data is shown in the leaves to the left of the stem. Both sets of data are arranged from smallest to largest, with the values getting larger as they move away from the stem.

For example, here is a back-to-back stem and leaf plot showing the test results of students in two different classes.

Class 1 is shown on the right of the stem and class 2 is shown on the left of the stem.

The data is always written from smallest to largest as we move away from the stem. The data for class 1 gets larger as we move right and the data for class 2 gets larger as we move left.

The key helps to explain the data values on each side of the back-to-back stem and leaf plot.

For example, reading the data for class 1: |1|2 means 12.

Read the digits backwards on the left side of a back-to-back stem and leaf diagram. For example, 0|2| means 20.

Here is an example of a back-to-back stem and leaf plot with decimals.

The data for boys is written on the right of the stem, getting larger from left to right.

The data for girls is written on the left of the stem, getting larger from right to left.

For example, |1.5|6 means 1.56 m.

For example, 2|1.6| means 1.62 m. Again, we read backwards when reading the left leaves on a back-to-back stem and leaf plot.

Stem and Leaf Plot with Hundreds

To represent numbers in the hundreds on a stem and leaf plot, write the hundreds and tens digits in the stem and write the ones digit in the leaf.

For example, 12|4 means 124.

Stem and Leaf Plot with Thousands

To represent numbers with thousands on a stem and leaf plot, write the thousands, hundreds and tens digits in the stem and write the ones digit in the leaf.

For example, 211|8 means 2118.

Stem and Leaf Plots with Decimals

To write decimal numbers in a stem and leaf plot, write the final decimal digit in the leaf. Write the preceding digits in the stem. The decimal point may also be included in the stem. However, if the numbers only have one decimal place, the decimal point is not included.

Stem and Leaf Plots with 1 Decimal Place

To write decimal numbers with 1 decimal place on a stem and leaf plot, write the whole number digits in the stem and the single decimal digit in the leaf.

For example, 5|7 means 5.7. The line between the stem and the leaf acts as the decimal point.

Stem and Leaf Plots with Multiple Decimal Places

To represent a number with multiple decimal places on a stem and leaf plot, write the final digit of each number in the leaf part of the plot. Write all preceding digits including the decimal point in the stem.

For example, 1.5|6 means 1.56.

Now try our lesson on How to Read Coordinates where we learn how to read coordinates on a graph.

Write coordinates in brackets.

Write the x coordinate first, then a comma, then the y coordinate.

• A pair of coordinates are two numbers that tell us the location of a point on a grid.
• Coordinates are two numbers written in between brackets, separated by a comma.
• The first number is called the x-coordinate and tells us how far to the right our point is.
• The second number is called the y-coordinate and tells us how far up our point is.
• To help us remember the order we can say ‘along the corridor, up the stairs’.
• This helps us remember that the first number tells us how far along (right) and the second number tells us how far up.
• Coordinates must be written in this order.

• The coordinates of this point are (8, 5).
• The first number tells us how far to the right the point is.
• The 8 tells us the point is 8 right.
• The second number tells us how far up the point is.
• To 5 tells us the point is 5 up.

The first number tells us how far to the right the point is and the second number tells us how far up the point is.

If the first number is negative, the point is to the left. If the second number is negative, the point is downwards.

• A positive x coordinate tells us that the point is to the right.
• A negative x coordinate indicates that the point is to the left.
• A positive y coordinate indicates that the point is upwards.
• A negative y coordinate indicates that the point is downwards.
• The coordinate pair (-10, -4) has an x coordinate of -10 and a y coordinate of -4.
• The point is found 10 left and 4 down.

How to Read Coordinates on a Grid: Video Lesson

Coordinates are 2 numbers written in brackets, separated by a comma. The first number is the x coordinate which when positive, indicates how far right the point is and when negative, indicates how far left the point is. The second number is the y coordinate which when positive, indicates how far up the point is and when negative, indicates how far down the point is.

The order in which coordinates are written matters. The first number is the x coordinate and the second number is the y coordinate.

For example, the coordinate (4, 9) has an x coordinate of 4 and a y coordinate of 9.

The x coordinate is 4. This is a positive number and so, it means to move 4 places right. This means we move 4 places right from where the axes cross over at (0, 0).

The y coordinate is 9. This means that after moving 4 places right, we move 9 places up. We know that we move upwards because 9 is a positive number.

To remember the order that the first number tells us to go right and the second number tells us to go up, we use the phrase ‘along the corridor, up the stairs’.

The ‘along the corridor’ represents us going along (to the right) and the ‘up the stairs’ represents going up.

This phrase is commonly used in schools when teaching coordinates because it helps to avoid the most common mistake of writing the two numbers in the wrong order.

To read coordinates from a grid, compare the position of the coordinate to the origin where the axes meet. First write the number of places the coordinate is to the right of the y-axis. If it is to the left, this number is negative. Now write a comma and then write the number of places the coordinate is above the x-axis. If it is below, this number is negative.

Coordinates are always measured from the origin. The origin is where the x and y axes cross over at the centre of the grid.

We compare the position of the coordinate to the origin to read the coordinate.

• Right is positive.
• Left is negative.
• Up is positive.
• Down is negative.

The point is 10 left of the origin. Since it is left, we write down -10. Negative x coordinates mean left.

The point is 4 below the origin. Since it is down, we write -4. Negative y coordinates mean down.

The coordinates are written (-10, -4). The numbers must be written in this order.

Write the x coordinate first, which is how far left or right the coordinate is. Then write the y coordinate second, which is how far up or down the coordinate is.

How to Plot Coordinates

To plot coordinates:

1. Read the first coordinate before the comma.
2. Along the x-axis, move this many places right if it is positive or left if it is negative.
3. Read the second coordinate after the comma.
4. Move this many places up if the number is positive or down if it is negative.
5. Draw a cross at this position.

For example, plot (-8, 2).

The first number, the x coordinate, is -8. It is negative, so we move 8 places left.

Now we read the second number, the y coordinate, which is 2. It is positive so we move 2 places up.

For example, plot the coordinate (6, -6).

The x coordinate is positive, so we move 6 places right.

The y coordinate is negative, so we move 6 places down.

For example, plot (3, 5).

The x coordinate is 3, so we move 3 places right.

The y coordinate is 5, so we move 5 places up.

The Order of Coordinates Matters

The order in which coordinates are written matters. The x coordinate must be written first and the y coordinate must be written second. The x coordinate describes the horizontal position and the y coordinate describes the vertical position. If the x and y coordinates are written in the wrong order, the coordinate will be describing the wrong position.

For example:

The coordinate (1, 2) means 1 to the right and 2 up.

The coordinate (2, 1) means 2 to the right and 1 up.

We can see that switching the order of the coordinates results in the point being located in a different position.

When learning how to write coordinates, the most common mistake is to write the x and y coordinates in the wrong order.

Use the phrase, ‘along the corridor and up the stairs’ to remember that the first coordinate describes how far along the point is and the second coordinate describes how far up or down the point is.

Reading Coordinates with Fractions and Decimals

Coordinates can be located at any point on the grid, including inside the squares of the grid. If a coordinate is located directly between two whole numbers, it will contain fractions of one half. If the coordinate is not directly in between two whole numbers, use the scale to estimate the location as a decimal.

For example, here is the coordinate (4   1/2, 3   1/2).

The coordinate is located directly between 4 and 5 on the x axis and so has an x coordinate of 4   1/2.

The coordinate is located directly between 3 and 4 on the y axis and so has a y coordinate of 3   1/2.

Here is an example of a coordinate that is not in the centre of a grid and so, it is nearer to one number than the other.

Here we estimate its position and we can write it as a decimal.

The coordinate is very close to 2 on the x axis but it is not quite on 2. It is at 1.9.

The coordinate is very close to 1 on the y axis but it is just above 1. It is at 1.1.

To read coordinates in 3D, read the distance travelled in the direction of the x axis, y axis and z axis in that order. There are 3 numbers that make up a set of coordinates in 3D and they are always written in the form (x, y, z).

For example, Here is the point (1, 3, 2).

The point has an x coordinate of 1, a y coordinate of 3 and a z coordinate of 2.

Here is the 3D coordinate of (2, 0, 5).

The point has an x coordinate of 2, a y coordinate of 0 and a z coordinate of 5.

Now try our lesson on Reflecting Shapes where we learn how to reflect a shape on a grid.

All About the Bridging to 10 Strategy

All About the Bridging to 10 Strategy

The bridging to 10 strategy is a method used to add two numbers. First add part of the number to get to 10 and then add the remainder.

• Here we will add 6 to 9.
• 6 is equal to 1 plus 5.
• We add 1 to 9 to get to 10.
• We then add the remainder of 5 to 10 to get to 15.
• We calculated 9 + 1 = 10 and then 10 + 5 = 15.
• In total we have added 1 and 5, which is the same as adding 6.
• The bridging to 10 strategy is useful because it makes addition easier.
• This is because it uses number bonds to 10, which are commonly memorised.
• It is also easy to add a number to 10.

• We add 18 + 8 using the bridging to 10 strategy.
• The next ten after 18 is 20.
• We first add 2 to 18 to make 20.
• We have added 2 and we need to add 6 more in order to add 8 in total.
• We add 6 more to 20 to make 26.
• 18 + 8 = 26.

Supporting Lessons

Bridging to 10

What is Bridging to 10?

Bridging to 10 is a method of adding two numbers that have an answer larger than 10. Count up to 10 and then add on the remainder. For example, to work out 7 + 4, firstly do 7 + 3 = 10 and then add the remainder of 1 to make 11.

The bridging to 10 strategy is often referred to as the bridging through 10 strategy or simply the bridging 10 strategy.

The bridging 10 strategy is a useful strategy for mental addition.

How to Teach Bridging to 10

When learning the bridging to 10 strategy, it is first necessary to have a strong understanding of the number bonds to 10. Number lines help to teach bridging to 10 because they can be used to count up to the next ten. Furthermore, tens frames and counters can be used to help count up to the next ten.

For example, the number line below shows the bridging to 10 of the addition 7 + 5.

We can count on by 3 to reach ten and then see that 2 more must be added to add 5 in total.

It helps to know the number bonds to 10 very well before learning the bridging to 10 method. This way we know that 3 must be added to 7 to make 10.

Tens frames can be used to help teach the idea of number bonds to ten and they can also be used to teach the bridging to 10 process.

Tens frames allow us to easily see how many more must be added to a number in order to make ten. This helps us with the first step of the bridging ten process.

To add two numbers by bridging to 10:

1. Find the number that when added to the larger of the two numbers makes the next multiple of ten.
2. Find the remainder by subtracting this same number from the smaller number.
3. Add this remainder onto the multiple of ten found in step 1.

For example, add 46 + 7 by bridging to 10.

Step 1. Find the number that when added to the larger of the two numbers makes the next multiple of ten

46 is the larger of the two numbers. We can add 4 to 46 to make the next multiple of ten which is 50.

Step 2. Find the remainder by subtracting this same number from the smaller number

We subtract 4 from 7 to make 3. This means that the remainder is 3.

Step 3. Add this remainder onto the multiple of ten found in step 1

The remainder is 3. We add this to the multiple of ten found in step 1, which was 50.

50 + 3 = 53.

Therefore 46 + 7 = 53.

We bridged to the next ten along which was 50. Since we added 4, we needed to add 3 more to add a total of 7.

Subtraction by Bridging to 10

To subtract by bridging to 10:

1. Subtract the ones digit of the larger number from the larger number to make a multiple of ten.
2. Find how much more must be subtracted in order to subtract the amount required.
3. Subtract this extra amount from the multiple of ten to obtain the answer to the subtraction.

For example, calculate 42 – 7 using the bridging to ten subtraction strategy.

Step 1. Subtract the ones digit of the larger number from the larger number to make a multiple of ten

We subtract 2 from 42 to obtain 40.

Step 2. Find how much more must be subtracted in order to subtract the amount required

We need to subtract 7 and we have already subtracted 2.

We need to subtract 5 more because 2 + 5 = 7.

Step 3. Subtract this extra amount from the multiple of ten to obtain the answer to the subtraction

We subtract 5 more from 40 to make 35.

Therefore 42 – 7 = 35.

We subtracted 2 to bridge ten at 40 and then subtracted 5 more to subtract a total of 7.

Now try our lesson on Subtraction on an Empty Number Line where we learn how to subtract a large number in chunks.

What are Cube Numbers?

What are Cube Numbers?

The first ten cube numbers are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000.

A cube number is the result of multiplying a number by itself twice. For example 23 = 2 Ã— 2 Ã— 2 = 8.

• When any whole number is multiplied by itself twice a cube number is formed.
• For example, 2 cubed means 2 Ã— 2 Ã— 2 which equals 8.
• Therefore we say that 8 is a cube number.
• ‘Cubing’ a number means to multiply a number by itself twice.
• We write a small 3 above the number to tell us to cube it like so: 23.
• Cube numbers are called this because cubing a side length of a cube gives us the volume of a cube.
• If a cube has a side length that is a whole number, its volume will be a cube number.

• To cube a number, multiply it by itself twice.
• 2 cubed is written as 23.
• 2 Ã— 2 Ã— 2 = 8 and so, 2 cubed is 8.
• 3 cubed is written as 33.
• 3 Ã— 3 Ã— 3 = 27 and so, 3 cubed is 27.
• 4 cubed is written as 43.
• 4 Ã— 4 Ã— 4 = 64 and so, 4 cubed is 64.

Supporting Lessons

Cube Numbers

What are Cube Numbers?

A cube number is formed when any whole number is multiplied by itself twice. The symbol for cubing a number is 3. For example, 2 cubed or 23 means 2 Ã— 2 Ã— 2 which equals 8. Therefore 8 is a cube number. Cube numbers are called this because the volume of a cube is found by cubing its side length.

Below shows a cube with side lengths of 2 cm.

The volume of the cube is found by multiplying a side length by itself twice.

2 Ã— 2 Ã— 2 = 8 and so the volume of the cube is 8 cm3.

This means that the overall cube is made up of 8 smaller 1 cm3 cubes.

8 is a cube number because it is formed by multiplying a whole number by itself twice.

2 Ã— 2 Ã— 2 = 8

We can write this more simply as 23 = 8.

The number is written to the power of 3, which is a shorter way of writing the full multiplication.

How to Find a Cube Number

To find a cube number, multiply any whole number by itself and then by itself again. The easiest way to do this is to do each multiplication separately. For example, 3 cubed is 3 Ã— 3 Ã— 3. The first multiplication is 3 Ã— 3 = 9 and then the second multiplication is 9 Ã— 3 = 27. Therefore the 3rd cube number is 27.

Here is another example of calculating 4 cubed.

43 means 4 Ã— 4 Ã— 4.

The first step is to multiply 4 Ã— 4 to get 16.

The next step is to multiply 16 by 4 again.

16 Ã— 4 = 64.

64 is the 4th cube number.

List of Cube Numbers

The first ten cube numbers are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000.

Here is a complete list of the first 100 cube numbers:

Number Calculation Cube Number
131 Ã— 1 Ã— 1 =1
232 Ã— 2 Ã— 2 =8
333 Ã— 3 Ã— 3 =27
434 Ã— 4 Ã— 4 =64
535 Ã— 5 Ã— 5 =125
636 Ã— 6 Ã— 6 =216
737 Ã— 7 Ã— 7 =343
838 Ã— 8 Ã— 8 =512
939 Ã— 9 Ã— 9 =729
10310 Ã— 10 Ã— 10 =1000
11311 Ã— 11 Ã— 11 =1331
12312 Ã— 12 Ã— 12 =1728
13313 Ã— 13 Ã— 13 =2197
14314 Ã— 14 Ã— 14 =2744
15315 Ã— 15 Ã— 15 =3375
16316 Ã— 16 Ã— 16 =4096
17317 Ã— 17 Ã— 17 =4913
18318 Ã— 18 Ã— 18 =5832
19319 Ã— 19 Ã— 19 =6859
20320 Ã— 20 Ã— 20 =8000
21321 Ã— 21 Ã— 21 =9261
22322 Ã— 22 Ã— 22 =10648
23323 Ã— 23 Ã— 23 =12167
24324 Ã— 24 Ã— 24 =13824
25325 Ã— 25 Ã— 25 =15625
26326 Ã— 26 Ã— 26 =17576
27327 Ã— 27 Ã— 27 =19683
28328 Ã— 28 Ã— 28 =21952
29329 Ã— 29 Ã— 29 =24389
30330 Ã— 30 Ã— 30 =27000
31331 Ã— 31 Ã— 31 =29791
32332 Ã— 32 Ã— 32 =32768
33333 Ã— 33 Ã— 33 =35937
34334 Ã— 34 Ã— 34 =39304
35335 Ã— 35 Ã— 35 =42875
36336 Ã— 36 Ã— 36 =46656
37337 Ã— 37 Ã— 37 =50653
38338 Ã— 38 Ã— 38 =54872
39339 Ã— 39 Ã— 39 =59319
40340 Ã— 40 Ã— 40 =64000
41341 Ã— 41 Ã— 41 =68921
42342 Ã— 42 Ã— 42 =74088
43343 Ã— 43 Ã— 43 =79507
44344 Ã— 44 Ã— 44 =85184
45345 Ã— 45 Ã— 45 =91125
46346 Ã— 46 Ã— 46 =97336
47347 Ã— 47 Ã— 47 =103823
48348 Ã— 48 Ã— 48 =110592
49349 Ã— 49 Ã— 49 =117649
50350 Ã— 50 Ã— 50 =125000
51351 Ã— 51 Ã— 51 =132651
52352 Ã— 52 Ã— 52 =140608
53353 Ã— 53 Ã— 53 =148877
54354 Ã— 54 Ã— 54 =157464
55355 Ã— 55 Ã— 55 =166375
56356 Ã— 56 Ã— 56 =175616
57357 Ã— 57 Ã— 57 =185193
58358 Ã— 58 Ã— 58 =195112
59359 Ã— 59 Ã— 59 =205379
60360 Ã— 60 Ã— 60 =216000
61361 Ã— 61 Ã— 61 =226981
62362 Ã— 62 Ã— 62 =238328
63363 Ã— 63 Ã— 63 =250047
64364 Ã— 64 Ã— 64 =262144
65365 Ã— 65 Ã— 65 =274625
66366 Ã— 66 Ã— 66 =287496
67367 Ã— 67 Ã— 67 =300763
68368 Ã— 68 Ã— 68 =314432
69369 Ã— 69 Ã— 69 =328509
70370 Ã— 70 Ã— 70 =343000
71371 Ã— 71 Ã— 71 =357911
72372 Ã— 72 Ã— 72 =373248
73373 Ã— 73 Ã— 73 =389017
74374 Ã— 74 Ã— 74 =405224
75375 Ã— 75 Ã— 75 =421875
76376 Ã— 76 Ã— 76 =438976
77377 Ã— 77 Ã— 77 =456533
78378 Ã— 78 Ã— 78 =474552
79379 Ã— 79 Ã— 79 =493039
80380 Ã— 80 Ã— 80 =512000
81381 Ã— 81 Ã— 81 =531441
82382 Ã— 82 Ã— 82 =551368
83383 Ã— 83 Ã— 83 =571787
84384 Ã— 84 Ã— 84 =592704
85385 Ã— 85 Ã— 85 =614125
86386 Ã— 86 Ã— 86 =636056
87387 Ã— 87 Ã— 87 =658503
88388 Ã— 88 Ã— 88 =681472
89389 Ã— 89 Ã— 89 =704969
90390 Ã— 90 Ã— 90 =729000
91391 Ã— 91 Ã— 91 =753571
92392 Ã— 92 Ã— 92 = 778688
93393 Ã— 93 Ã— 93 =804357
94394 Ã— 94 Ã— 94 =830584
95395 Ã— 95 Ã— 95 =857375
96396 Ã— 96 Ã— 96 =884736
97397 Ã— 97 Ã— 97 =912673
98398 Ã— 98 Ã— 98 =941192
99399 Ã— 99 Ã— 99 =970299
1003100 Ã— 100 Ã— 100 =1000000

Properties of Cube Numbers

Cube numbers have the following properties:

• An even number cubed is always even.
• An odd number cubed is always odd.
• A positive number cubed is always positive.
• A negative number cubed is always negative.
• Cubing a number ending in 0, 1, 4, 5, 6 or 9 will result in a number ending in this same digit.
• The sum of the cubes of the first n natural numbers is equal to the square of their sum.

Here are some examples to illustrate these properties of cube numbers.

An even number cubed is always even

For example, 2 is an even number. If we cube it we get 8, which is an even answer.

2 Ã— 2 Ã— 2 = 8

This property works because if any number is multiplied by an even number at least once, the result is even. Cubing an even number means that we must multiply by an even number.

An odd number cubed is always odd

For example, 3 is an odd number. If we cube it we get 27, which is an odd answer.

3 Ã— 3 Ã— 3 = 27

This property works because if two odd numbers are multiplied together, the result is always odd. To make an even number, at least one of the numbers being multiplied will need an even factor. However, when an odd number is cubed, we have odd Ã— odd Ã— odd and so, no factor of two appears in the final result.

A positive number cubed is always positive

For example, 10 is a positive number. If we cube it we get 1000, which is also positive.

10 Ã— 10 Ã— 10 = 1000

This property works because a negative number can only be made from another negative. If we only multiply positive numbers, the result must be positive.

A negative number cubed is always a negative

For example, (-2) Ã— (-2) Ã— (-2) = -8.

When we cube a number, we multiply it by itself twice. If a negative number is cubed, we have three negative numbers multiplied together.

When three negative numbers are multiplied together, the result is always negative.

(-2) Ã— (-2) = +4 and then 4 Ã— (-2) = -8.

Cubing a number ending in 0, 1, 4, 5, 6 or 9 will result in a number ending in this same digit

For example:

103 = 1000. Both numbers end in 0.

213 = 9261. Both numbers end in 1.

43 = 64. Both Numbers end in 4.

153 = 3375. Both numbers end in 5.

663 = 287496. Both numbers end in 6.

193 = 6859. Both numbers end in 9.

The sum of the cubes of the first n natural numbers is equal to the square of their sum

13 + 23 + 33 + … + n3 = (1 + 2 + 3 + … + n)2.

For example, for an n of 5:

13 + 23 + 33 + 43 + 53 = 225.

(1 + 2 + 3 + 4 + 5)2 = 225.

Cubing the consecutive numbers and then adding them up gives the same result as adding the numbers up and then squaring them.

For example, for an n of 10:

13 + 23 + 33 + 43 + 53 + 63 + 73 + 83 + 93 + 103 = 3025.

(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)2 = 3025.

Cube Numbers That Are Also Square Numbers

The cube numbers that are also square numbers are found by raising whole numbers to the power of 6. For example, 16 = 1, 26 = 64 and 36 = 729. 1 is 12 and 13, 64 is 82 and 43 and 729 is 272 and 93.

13 = 1 = 12 23 = 8 33 = 27 43 = 64 = 82 53 = 125 63 = 216 73 = 343 83 = 512 93 = 729 = 272 103 = 1000

1, 64 and 729 are cube numbers and also square numbers.

Further cube numbers that are also square numbers are found by raising any integer to the power of 6:

16 = 1 26 = 64 36 = 729 46 = 4096 56 = 15625 66 = 46656 76 = 117649 86 = 262144 96 = 531441 106 = 1000000

Now try our lesson on Finding Prime Numbers to 100 where we learn how to find prime numbers.

Cube Numbers Activity

Cuisenaire rods are physical mathematical learning supports in which each colour rod represents a different number.

• The lengths of each colour Cuisenaire rod is shown in the chart above. The white rod is 1 cm long.
• Each colour rod is worth a different value.
• Cuisenaire rods are useful for students to build their number sense as the rods allow for easy comparison between each length.
• The rods can also be used to build understanding of the concepts of addition, subtraction, multiplication, division, fractions, decimals and ratio.

• Cuisenaire rods can be used to teach multiplication as a repeated addition process.
• We can line up Cuisenaire rods to show that many smaller quantities are equivalent to one larger quantity.
• For example we can see that five lots of the 2 rods are equivalent in length to one 10 rod.
• This can help to reinforce the conceptual understanding of 2 Ã— 5 = 10 and other multiplication facts.

Supporting Lessons

Cuisenaire Rods

What are Cuisenaire Rods?

Cuisenaire rods are coloured plastic bars of different lengths, which are used to represent different number sizes. These rods are used to demonstrate the comparative sizes of numbers and can be useful in teaching basic addition, subtraction, multiplication and division. It is also possible to use Cuisenaire rods to demonstrate equivalent fractions, decimal numbers and ratio.

Here is a colour chart showing the different Cuisenaire rod values.

Cuisenaire rods were invented by Belgian schoolteacher Emile-Georges Cuisenaire in the 1950s.

They are often used in primary schools for teaching number facts and therefore are commonly used with children between the ages of 4-7. However, they are often used with children older than this who struggle with mathematics as they can support their understanding of number size. Cuisenaire rods can also be used to teach older children between 11-16 concepts such as ratios and fractions.

How to Introduce Cuisenaire Rods

Cuisenaire rods can be introduced by asking for the rods to be placed in order of size and then asking if any smaller rods can be combined to make the same length as the larger rods. Then ask if further combinations of rods that are of equal lengths can be found. Can we make any number using only white or red rods? These activities will help build the idea of equivalences and familiarise children with the rod values.

Showing basic equivalences can be helpful such as this example of 2 + 3 = 5. Can your child find further combinations of rods that are equal in size?

Further introductory Cuisenaire rod ideas are:

• Can you make a combination of rods the same length as the blue rod using exactly 3 rods?
• Can you make a rod the same length as the orange rod using only red rods?
• How about using only green rods?
• What about using only red and green rods?
• What is larger, a black plus a white or a brown plus a purple?

Games that ask children to create equivalent lengths and to play with the size of different rods are useful for introducing Cuisenaire rods.

Why Use Cuisenaire Rods

Cuisenaire rods are used because they are a simple and hands-on way to compare number sizes. They help to engage children in mathematics as they are brightly coloured and easy to stack together. Cuisenaire rods also have a number of purposes across many different branches of mathematics which makes them a versatile tool.

Here are some of the advantages and disadvantages of teaching with Cuisenaire rods.

They provide tactile learning for students with maths difficulties You cannot count up in ones on each block
Helps to build comparative number sense The rods can be limited to teaching smaller numbers
Different coloured rods for each size Lengths are not written on the rods
Can be used in a variety of games/settings Cannot represent all fractions easily
Simple to use Potentially limited in challenge
Allows for trial and error without fear of making mistakes Limited feedback on learning
Time to explore numbers without rushing
Can build stronger understanding of numbers using visual learning

What are the Values of the Cuisenaire Rods?

Cuisenaire rods always have the same fixed values:

Cuisenaire Rod Colour Colour Code Value Length
White w 1 1 cm
Red r 2 2 cm
Lime Green g 3 3 cm
Purple p 4 4 cm
Yellow y 5 5 cm
Dark Green d 6 6 cm
Black b 7 7 cm
Tan (Brown) t 8 8 cm
Blue B 9 9 cm
Orange o 10 10 cm

How to Use Cuisenaire Rods to Teach Addition

To teach addition using Cuisenaire rods, combine two smaller rods end to end and place them alongside a larger rod that is the same length as the two smaller rods combined. For example 2 white rods are the same length as 1 red rod. We can say that 1 + 1 = 2 because they are the same length. This can then be extended to larger numbers by combining two or more larger rods.

For example, the red plus green rods are the same length as the yellow rod.

Since the red = 2, the green = 3 and the yellow = 5, we can see that 2 + 3 = 5.

Here is a larger example of 4 + 6 = 10 using the purple, dark green and orange rods.

Cuisenaire rods are great for teaching the number bonds to 10. These are the pairs of numbers that add to make 10.

Simply ask your child to find and list as many different combinations of rods that add to make the same length as the orange 10 rod.

How to Use Cuisenaire Rods to Teach Subtraction

To teach subtraction using Cuisenaire rods, find a combination of two smaller rods that are the same length as one larger rod. Place the two smaller rods alongside the larger rod to show that they are the same size. Then remove the rod that you are subtracting and the remaining rod is the answer. Cuisenaire rods help to reinforce that the subtraction gives us the difference between two numbers.

For example, here is 7 – 4 = 3.

The black rod is worth 7, the purple is worth 4 and the lime green is worth 3.

To show the subtraction of 4, simply remove the purple 4 rod. The remaining green rod is the answer. This shows us that 3 is the difference between 7 and 4.

Here is another example of subtraction with Cuisenaire rods. We have 9 – 2 = 7.

The blue is worth 9, the red is worth 2 and the black is worth 7.

Removing the red 2 rod leaves us with the red 7 rod. 7 is the difference between 9 and 2.

How to Use Cuisenaire Rods to Teach Multiplication

Cuisenaire rods can be used to teach multiplication by combining rods of the same size end-to-end to make a larger value. The number of rods combined together tell us how many lots of the base number we have. For example, 2 Ã— 5 = 10 can be shown by lining up 5 lots of the red size 2 rods alongside an orange size 10 rod.

We can see that 5 lots of 2 is equal to 10.

Here is an introductory example of 1 Ã— 4 = 4.

Each white Cuisenaire rod is worth 1 and the purple Cuisenaire rod is worth 4. We can see that 4 lots of 1 is the same as 4.

Cuisenaire rods can be used to build a conceptual understanding of multiplication as repeated addition.

How to Use Cuisenaire Rods to Teach Division

Cuisenaire rods can be used to teach division by first combining rods of the same size end-to-end to form a rod of a larger size. The smaller rods can then be counted to see how many go into the larger rod. The number of smaller rods is the answer to the division. For example, a blue size 9 rod can be made from 3 lime green size 3 rods. This demonstrates the division of 9 Ã· 3 = 3.

There are 3 lots of the smaller lime green rods and so, the answer to the division is 3.

Here is another example of division using Cuisenaire rods. We have 6 Ã· 1 = 6.

The dark green rod is worth 6. The white rod is worth 1.

There are 6 white rods and so, 6 Ã· 1 = 6.

How to Use Cuisenaire Rods to Teach Fractions

To teach fractions using Cuisenaire rods, a smaller rod can be used to represent the numerator and the larger rod can be used to represent the denominator. For example a size 2 red rod and a size 6 dark green rod represent the fraction 2/6. We can see that the red rod is one third of the size of the dark green rod and so, 2/6 is the same as 1/3.

The number of times the smaller rod fits into the larger rod tells us the simplified fraction.

Cuisenaire rods are useful to build basic fraction understanding, such as comparing a size 5 yellow with a size 10 orange rod can easily show that 5 is half of ten.

How to Use Cuisenaire Rods to Teach Decimals

Cuisenaire rods can be useful tools for explaining the comparative sizes of decimals. For example, labelling the size 10 orange rod as worth 1, then the size 1 white rod will be one tenth of this, which is 0.1. Then the red size 2 rod is worth 0.2 and so on. Decimals can be taught by combining Cuisenaire rods to make further decimal numbers.

Here the orange rod is worth one whole. This means that every white size 1 rod is worth 0.1.

We can combine 3 white rods to make 0.3.

Alternatively, two red size 2 rods would be worth 0.4 because this is the same length as 4 of the 0.1 rods.

Cuisenaire rods are very useful when teaching the comparative sizes of decimals. In the diagram above we can easily see that 0.4 is just under half of the size of 1 whole.

We can also represent decimals in other ways if necessary. For example, this time the yellow size 5 rod is worth one whole.

This means that each size 1 white rod is one fifth, or 0.2.

Every time we add a white rod, we count up in 0.2s.

How to Use Cuisenaire Rods to Teach Ratio

Due to the ease in which Cuisenaire rod sizes can be compared, they are a very useful tool for teaching ratios. By allocating a value to the smallest white rod, larger values can be demonstrated. For example, if the white size 1 rod is worth 5, then the red size 2 rod is worth 10.

The red rod is always double the value of the white rod, no matter what the white rod is worth.

Here is another example of teaching ratio using Cuisenaire rods. Here the lime green size 3 rod is worth 30.

Because 3 white rods make 1 lime green rod, the white rod must be worth 10.

We can then see that the yellow rod must be worth 50 because we have 5 lots of the white rods, which are worth 10 each.

Here we have a purple rod worth 80 and we want to know the value of the red rod. We first see that 4 white rods go into the purple rod, so each white rod must be worth 20.

The red rod is always double the white rod and so, it is worth 40.

Cuisenaire Rod Activities

Here are a list of short Cuisenaire rod activities:

• Put the Cuisenaire rods in order of size from smallest to largest to make a ‘staircase’.
• Count how many white rods are needed to make the other size rods.
• Find which colour rods can be make using only the red rods.
• Find as many combinations of two smaller rods which are the same length as one larger rod.
• Find as many combinations of three smaller rods which are the same length as one larger rod.
• Find as many different pairs of rods that have a difference in size of 1.
• Find a pair of rods that have a difference of exactly 6.
• Use the rods to find and list all different combinations of rods that add to make 10.
• Cuisenaire rod snap – take it in turns to draw a different rod from your pile. If the two rods add to make 10, the person who says snap first wins the rods.

Now try our lesson on Place Value with Base Ten Blocks where we learn how to use base ten blocks.

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