## How to Make and Read Back-to-Back Stem-and-Leaf Plots

How to Make and Read Back-to-Back Stem-and-Leaf Plots

Back-to-back stem-and-leaf diagrams show two sets of data on each side of the central stem.

Only the final digit of each number is listed in the leaf.

All of the preceding digits are listed in the stem.

• The data for class 1 is written on the right of the stem and class 2 is written on the left of the stem.
• Only the last number is written in the leaves and the numbers get larger as we move away from the stem.
• In this case, the stem tells us the tens digit of the numbers in the leaves.
• The number in the stem tells us the tens digit for all of the numbers that follow in the leaf.
• The numbers in class 2 get larger as we go leftwards, away from the stem.

• The top row of data for class 2 is: 3, 6 and 8.
• The second row of data for class 2 is: 11, 12, 13, 14, 14, 17.
• The third row of data for class 2 is: 20 and 27.
• The fourth row of data for class 2 is: 32.

• The top row of data for class 1 is: 5 and 9.
• The second row of data for class 1 is: 10, 10, 12, 12, 15, 15, 16, 19.
• The third row of data for class 1 is: 23.
• The fourth row of data for class 1 is: 31.

• Only the final digit is listed in the leaves to represent each number.
• For example, in the girls data, 1.39 is listed by writing a 9 next to the 1.3 in the top row.
• In the second row, the girls have heights of 1.46, 1.47 and 1.46 listed. The boys have a height of 1.49.
• In the third row, the girls have heights of 1.51 and 1.53. The boys have heights of 1.56 and 1.58.

• In the fourth row, the girls have heights of 1.60, 1.62 and 1.62. The boys have heights of 1.60, 1.61, 1.64, 1.66 and 1.66.
• No girls or boys have any heights starting with 1.7.
• Finally, there is one boy who is 1.82 metres tall

# Back-to-Back Stem-and-Leaf Plots

## What is a Back-to-Back Stem-and-Leaf Plot?

A back-to-back stem-and-leaf plot is used to compare two data distributions. The ‘stem’ is shared by both sets of data which are listed in the ‘leaves’ to the left and right of the stem respectively. The data is ordered so that the larger values are further from the stem.

The back-to-back stem-and-leaf plot below shows the data comparing the test scores of two classes.

In this example, the stem tells us the tens digit of each number and the numbers in the leaves tell us the ones digit.

For example, the 7 shown in line with the 2 in the stem represents 27.

The 5 shown in line with the 1 stem represents 15.

On a back-to-back stem and leaf plot, a key is drawn to help explain how to read the diagram. The key on a stem-and-leaf plot illustrates an example.

For example, 7 |1| means 17. The digit of 7 is in the left leaf and so, it is read backwards.

|1| 2 means 12. Since the 2 is in the right leaf, the number is read forwards.

## How to Draw a Back-to-Back Stem-and-Leaf Diagram

To make a back-to-back stem-and-leaf diagram:

1. Put the numbers in each data set in order from smallest to largest.
2. All apart from the final digit of each number is listed so that they appear in the central stem once.
3. The final digits in data set 1 are listed to the left of the stem, getting larger as they get further from the stem.
4. The final digits in data set 2 are listed to the right of the stem, getting larger as they get further from the stem.
1. For example, construct a back-to-back stem-and-leaf plot for the data shown. The stem and leaf plot shows the number of goals scored by each team over the season.

Team 1: 2, 9, 35, 21, 20, 10, 6, 13, 24, 31, 3, 1 Team 2: 5, 6, 19, 12, 15, 22, 30, 28, 21, 14, 17, 8

Step 1: Put the numbers in each data set in order from smallest to largest

Writing the numbers in order we obtain:

Team 1: 1, 2, 3, 6, 9, 10, 13, 20, 21, 24, 31, 35 Team 2: 5, 6, 8, 12, 14, 15, 17, 19, 21, 22, 28, 30

Step 2: All apart from the final digit of each number is listed so that they appear in the central stem once

The digits listed in the stem will be the tens digits of each number. The ones digits of each number will go in the leaves.

The smallest value to list is a 1. It has zero tens and so, a 0 is written in the stem.

The largest values to list is a 35. It has 3 tens and so, the numbers in the stem will be listed from 0 up to 3.

Step 3: The final digits in data set 1 are listed to the left of the stem, getting larger as they get further from the stem

For example, entering the numbers with zero tens from team 1: we enter 1, 2, 3, 6 and 9.

These numbers are entered as 9 6 3 2 1 |0|.

The 0 in the stem shows that there is nothing in the tens column.

The numbers are listed backwards because they are listed so that the larger numbers are furthest from the stem.

The numbers of 10 and 13 are listed as 3 0 |1|.

Both numbers have a 1 in the tens column, so the stem has a 1 in it.

The numbers of 20, 21 and 24 are listed as 4 1 0 |2|.

The numbers of 31 and 35 are listed as 5 1 |3|.

Step 4: The final digits in data set 2 are listed to the right of the stem, getting larger as they get further from the stem

5, 6 and 8 are listed as |0| 5 6 8.

12, 14, 15, 17 and 19 are listed as |1| 2 4 5 7 9.

21, 22 and 28 are listed as |2| 1 2 8.

30 is listed as |3| 0.

## Back-to-Back Stem-and-Leaf Plot with 3-Digit Numbers

When constructing a stem-and-leaf plot with 3-digit numbers, the hundreds and tens digits will go in the stem and the ones digit of each number will go in the leaves.

The back-to-back stem-and-lead plot shown below represents the heights (in centimetres) of two samples of plants.

Listing the values in sample 1:

101, 102, 102 and 108 are listed as 8 2 2 1 |10|.

This is because they all have a 1 and a 0 in their hundreds and tens columns respectively. Only the ones digit of each number is listed.

115 is listed as 5 |11|.

123, 125 and 129 are listed as 9 5 3 |12|.

134 and 137 are listed as 7 4 |13|.

140 and 145 are listed as 5 0 |14|.

Listing the values in sample 2:

103, 104, 108 and 108 are listed as |10| 3 4 8 8.

110 and 117 are listed as |11| 0 7.

123, 124, 124, 127 and 128 are listed as |12| 3 4 4 7 8.

132 is listed as |13| 2.

There are no plants in sample 2 with a 1 in the hundreds column and a 4 in the tens column.

## Back-to-Back Stem-and-Leaf Plot with Decimals

Only the final decimal digit is listed in a stem-and-leaf plot. With decimal numbers, all other preceding digits are listed in the stem. The decimal point may also be listed in the stem.

For example, the following stem-and-leaf plot displays the heights of boys and girls in a class.

Listing the boy’s data:

1.49 is listed as |1.4| 9.

1.56 and 1.58 are listed as |1.5| 6 8.

1.60, 1.61, 1.64, 1.66 and 1.66 are listed as |1.6| 0 1 4 6 6.

1.82 is listed as |1.8| 2.

Listing the girl’s data:

1.39 is listed as 9 |1.3|.

1.46, 1.47 and 1.47 are listed as 7 7 6 |1.4|.

1.51 and 1.53 are listed as 3 1 |1.5|.

1.60, 1.62 and 1.63 are listed as 3 2 0 |1.6|.

## How to Use a Back-to-Back Stem-and-Lead Plot to Compare Data Sets

Back-to-back stem-and-leaf plots are used to compare two data sets. They are a visual display of the shape of the data, so that skewness, range and modal values can more easily be seen. The sizes of the values within the two data sets can easily be compared by looking at their general position on the diagram.

For example, the back-to-back stem-and-leaf plot below shows the heights of boys and girls in a class. The mode, median, range and skewness of the data can easily be found from the diagram.

The mode from a stem-and-leaf plot is found by looking for the digit that is repeated the most in a particular row.

The mode for the girl’s data is seen to be 1.47 m. This is because the only digit that is repeated in this data set is a 7 in the |1.4| stem.

The mode for the boy’s data is 1.66 m. This is because the only digit that is repeated in this data set is a 6 in the |1.6| stem.

The mode is a type of average and since the boys have a larger mode, we can interpret this as the boys having a larger height on average.

The median is found in the middle of each data set on a stem-and-leaf plot.

There are 9 data values in each set of data. By crossing off the smallest 4 data values and the largest 4 data values, the middle value will be left. This is the median.

In the girl’s data, the values of 1.39, 1.46, 1.47 and 1.47 are crossed off at the bottom, along with 1.63, 1.62, 1.60 and 1.53 at the top.

This leaves 1.51 metres as the median height for the girls.

In the boy’s data, the values of 1.49, 1.56, 1.58 and 1.60 are crossed off at the bottom, along with 1.82, 1.66, 1.66 and 1.64 at the top.

This leaves 1.61 metres as the median height for boys.

The median is a type of average and so, we can say that the boys are taller on average.

The range is found on a stem-and-lead plot by subtracting the first value from the last value

The first value in the girl’s data is 1.39 m. The last value is 1.63 m.

The range of the girl’s data is 1.63 – 1.39 = 0.24 m.

The first value in the boy’s data is 1.49 m. The last value is 1.82 m.

The range of the boy’s data is 1.82 – 1.49 = 0.33.

The range describes the spread of the data. Since the boys have a larger range, the boy’s heights are more spread than the girls.

The skewness of the data on a stem-and-leaf plot can be found by drawing around the peaks of each row.

The skew of the data can be seen below.

The girl’s data forms two peaks and therefore we can call it bimodal.

The boy’s data starts with a slow climb and most of the data is bunched at the bottom. This is called negatively skewed data.

In the opposite case, where most of the data is bunched at the top, this would be called positively skewed data.

A stem-and-leaf plot easily shows the positions of the data values.

It is easy to see that the boy’s values are typically lower down in the diagram than the girl’s values.

This tells us that the boy’s values are typically found in the larger numbers on the diagram and so, it is easy to see that boys are taller than girls on average in this example.

Now try our lesson on Probability with Spinners where we learn how to understand probability with spinners.

## How to Solve Equations by Inspection

How to Solve Equations by Inspection

To solve an equation by inspection, just think what number must be placed there to obtain the correct answer.

• Solving an equation means to find out the value of the variable (letter) which makes the equation equal to the value given.
• In the equation 𝑥 + 4 = 8, we think of the number that makes 8 when we add 4 to it. Since 4 + 4 = 8, 𝑥 must be equal to 4.
• In the equation 𝑥 – 3 = 4, we think of the number that makes 4 when we take away 3 from it. Since 7 – 3 = 4, 𝑥 must be equal to 7.
• In the equation 9 – 𝑥 = 8, we think of the number that we subtract from 9 to make 8. Since 9 – 1 = 8, 𝑥 must be equal to 1.
• In the equation 2𝑥 = 6, we think of a number that when doubled makes 6. Since 2 × 3 = 6, 𝑥 must be equal to 3.
• Equations can be solved by inspection by trialling different numbers until the correct answer is obtained.

• In the equation 2𝑥 + 1 = 9, we multiply 𝑥 by 2 and then add 1 to make an answer of 9.
• We think what number makes 9 when we double it and then add 1.
• 4 doubled is 8 and 8 + 1 = 9.
• Therefore 𝑥 = 4.

• Equations can be solved by trial and error by substituting values into the equation until the answer is reached.
• To solve 2𝑥 + 1 = 9, we substitute values of 𝑥 into the equation until we obtain an answer of 9.
• Starting with 𝑥 = 1, we obtain an answer of 3. Since 3 is smaller than 9, we need to try a larger value of 𝑥
• If 𝑥 = 2, 2𝑥 + 1 = 5.

• If 𝑥 = 3, 2𝑥 + 1 = 7.
• If 𝑥 = 4, 2𝑥 + 1 = 9.
• Since we have obtained a value of 9 when 𝑥 = 4, our solution is 𝑥 = 4.

Supporting Lessons

# Solving Equations by Inspection

## What is Solving by Inspection?

Solving an equation by inspection is a method used to obtain a solution without using algebra. The numbers in the equation are considered so that an estimate of the solution can be made. This guess is substituted into the equation to obtain an answer which is compared to the actual answer. Different values can be trialled until a solution is found.

For example, solve 𝑥 + 2 = 6 by inspection.

We think of the number that makes an answer of 6 when we add 2 to it.

We know that 4 is 2 less than 6 and so, 4 + 2 = 6.

Therefore 𝑥 must be equal to 4.

## How to Solve an Equation by Inspection

To solve an equation by inspection, use the existing numbers in the equation to make a guess at the solution. Substitute this value into the equation and compare the result to the actual answer. If the answer is too small, a larger estimate should be trialled. If the answer is too large, a smaller answer should be trialled.

For example, solve 5𝑥-2=28.

We know that 𝑥 will be multiplied by 5 and then 2 will be subtracted to get an answer of 28.

Since 𝑥 is multiplied by 5, we look for a number that when multiplied by 5 gives us a result close to 28.

We know that 5 × 6 = 30.

5 × 6 – 2 = 28 and so, 𝑥 = 6.

Here is another example.

Solve 2𝑥 + 3 = 19 by inspection.

This time we will substitute values into the equation with the goal of finding an answer of 19.

Since we are multiplying 𝑥 by 2, we can think of a multiple of 2 that is close to the answer of 19.

2 × 10 = 20 and so, we can try this.

2 × 10 + 3= 23.

This answer is too large and so, we try a smaller value of 𝑥.

We can substitute 𝑥 = 7 so that 2 × 7 + 3 = 17.

This is too small so we try a larger value of 𝑥.

We try 𝑥 = 8, which gives us 2 × 8 + 3 = 19.

This gives us the correct answer and so, 𝑥 = 8.

Now try our lesson on Solving One-Step Equations where we learn how to solve simple equations algebraically.

## The Near Doubles Addition Strategy

Use a known doubles fact and add or subtract from this.

• The near doubles strategy uses the known doubles facts to assist with addition.
• Double one of the numbers in the addition and count on or back from this result.
• For example, 5 + 6 is one more than 5 + 5.
• Since 5 + 5 = 10, we know that 5 + 6 is one more than this.
• Therefore 5 + 6 = 11.
• To use the near doubles strategy, it helps to know the doubles facts below.

• To calculate 9 + 8, we use the fact that 9 + 9 = 18.
• 9 + 8 must be one less than 9 + 9.
• Therefore 9 + 8 = 17.

Supporting Lessons

#### Near Doubles Activity

Click on the near doubles addition flashcards below to practise them:

+
+

=

# The Near Doubles Strategy

## What are Near Doubles?

Near doubles are maths facts that are close to doubles maths facts. For example, 3+4 is a near double because it is one more than 3+3. Near doubles are typically one or two larger or smaller than doubles facts.

Since we know that 3 + 3 = 6, we can work out 3 + 4 since it will be one larger.

3 + 4 = 7.

## How to do Near Doubles Addition

To add numbers using the near doubles strategy, double one of the numbers in the addition. Then add or subtract the difference between the other number and the number that was doubled.

For example, add 6 + 7.

Doubling 6, we get 12.

Looking at the other number, 7 is one larger than 6 and so, we must add one more to our answer.

6 + 7 = 13.

5 + 7 is a near double as it is close to 5 + 5.

Using the known double fact that 5 + 5 = 10, 5 + 7 must be two more than this and so, 5 + 7 = 12.

Near doubles addition may be used when the numbers are below the doubles facts.

For example, 8 + 7 is close to the doubles fact 8 + 8 = 16. However, since 7 is one less than 8, we subtract one from the answer.

Therefore 8 + 7 = 15.

## Teaching the Near Doubles Strategy

When teaching near doubles, first teach the doubles facts. Knowing these is necessary for the near doubles method to be effective. Using tens frames can be an effective teaching strategy for showing the comparative size of near doubles.

The near doubles addition strategy uses the knowledge of doubles facts as a crutch. This is because doubles facts are generally easily memorised.

The doubles facts that should be learnt are:

• 1 + 1 = 2
• 2 + 2 = 4
• 3 + 3 = 6
• 4 + 4 = 8
• 5 + 5 = 10
• 6 + 6 = 12
• 7 + 7 = 14
• 8 + 8 = 16
• 9 + 9 = 18
• 10 + 10 = 20

Tens frames are useful for teaching the concept of a ‘near double’.

In the tens frame below, 3 + 3 is represented with pink counters.

It can be seen that 3 + 4 is shown with an extra green counter. The green counter is used to represent the difference needed.

This can help conceptualise the idea of finding a difference to a known doubles fact.

It would be beneficial to first put the pink counters in place to represent the double. Then calculate the value of the double before adding the extra counters and then counting on from the previous total.

For example, 4 + 4 = 8 can be shown.

An extra counter can be placed next to one of the fours to make 4 + 5 = 9.

The idea is not to count the whole total again but simply count on from the known doubles fact of 4 + 4 = 8.

Once the conceptual idea of a near double has been taught, it is beneficial to practise them using tasks that require recall such as flashcards.

## Near Doubles Strategy for Subtraction

The near doubles strategy can be used for subtraction when the number being subtracted is close to half of the larger number. Use a known doubles fact and correct the result afterwards. For example, 12 – 6 = 6 and so 12 – 7 requires a further one to be subtracted. Therefore 12 – 7 = 5.

For example, 15 – 8 can be calculated using the near doubles subtraction strategy.

We know that 16 – 8 = 8 because 8 doubled is 16.

15 – 8 must be one smaller than 16 – 8.

Therefore 15 – 8 = 7.

Now try our lesson on Number Bonds to Ten where we learn the pairs of numbers that add to make 10.

## The Partial Sums Method

The Partial Sums Method

Add each place value column separately and then add up the totals.

• The partial sums method involves breaking each number down into their hundreds, tens and ones and adding them separately.
• 341 is made up of 300 + 40 + 1.
• 216 is made up of 200 + 10 + 6.
• Adding the hundreds of each number: 300 + 200 = 500.
• Adding the tens of each number: 40 + 10 = 50.
• Adding the ones of each number: 1 + 6 = 7.
• Finally, we add these results together: 500 + 50 + 7 = 557.

• In 55 + 36 we have 50 + 5 and 30 + 6.
• Adding the tens: 50 + 30 = 80.
• Adding the ones: 5 + 6 = 11.
• Finally, adding these partial sums: 80 + 11 = 91.

• In 231 + 194 we have 200 + 30 + 1 and 100 + 90 + 4
• Adding the hundreds: 200 + 100 = 300.
• Adding the tens: 30 + 90 = 120.
• Adding the ones: 1 + 4 = 5.
• Finally, adding these partial sums: 300 + 120 + 5 = 425.

Supporting Lessons

# The Partial Sums Method

## What is the Partial Sums Method?

The partial sums method is an addition strategy in which the numbers are added together in parts according to their place value. For example, 635 + 312 is calculated by adding 600 + 300 = 900, 30 + 10 = 40 and 5 + 2 = 7. These partial sums are then added as 900 + 40 + 7 = 947.

600 + 300 = 900

30 + 10 = 40

5 + 2 = 7

900 + 40 + 7 = 947

## Partial Sums Method Examples

Here are some examples of using the partial sums method.

Addition Sum Sum of the Hundreds Sum of the Tens Sum of the Ones Adding the Partial Sums
346 + 523 300 + 500 = 800 40 + 20 = 60 6 + 3 = 9 800 + 60 + 9 = 869
105 + 382 100 + 300 = 400 0 + 80 = 80 5 + 2 = 7 400 + 80 + 7 = 487
645 + 115 600 + 100 = 700 40 + 10 = 50 5 + 5 = 10 700 + 50 + 10 = 760
361 + 155 300 + 100 = 400 60 + 50 = 110 1 + 5 = 6 400 + 110 + 6 = 516
269 + 363 200 + 300 = 500 60 + 60 = 120 9 + 3 = 12 500 + 120 + 12 = 632

## Subtraction with Partial Differences

The partial differences subtraction method involves subtracting each place value column individually and then finding the total of these results. Always subtract the smaller number from the larger number. If the first number is subtracted from the second number, the result will be negative.

Use the partial differences method to subtract 625 – 453.

625 is made up of 600 + 20 + 5.

453 is made up of 400 + 50 + 3.

• Subtracting the hundreds

600 – 400 = 200

• Subtracting the tens

20 – 50 = -30.

Alternatively, 50 – 20 = 30 but since the numbers were subtracted the other way around, we write -30.

• Subtracting the ones

5 – 3 = 2

• Finding the total

200 – 30 + 2 = 172.

Therefore 625 – 453 = 172.

Here is another example of using the partial differences method of subtraction.

Calculate 814 – 366.

• Subtracting the hundreds

800 – 300 = 500

• Subtracting the tens

We need to subtract the numbers in the opposite order and therefore, the result will be negative.

60 – 10 = -50.

• Subtracting the ones

Again, we need to subtract the numbers in the opposite order and therefore, the result will be negative.

6 – 4 = -2.

• Finding the total

500 – 50 – 2 = 448.

Therefore 814 – 366 = 448.

Now try our lesson on Addition Using the Compensation Method where we learn how to use the compensation method for adding two numbers.

## How to Find the Difference Between Two Numbers

How to Find the Difference Between Two Numbers

To find the difference between two numbers, subtract the smaller number from the larger number.

• To find the difference between two numbers, subtract the smaller number from the larger number.
• 5 – 3 = 2 and so, the difference between 5 and 3 is 2.
• Alternatively, the difference between two numbers can be found by counting on from the smaller number.
• Count on from the smaller number until you make the larger number. This amount is the difference.

• The difference between 7 and 2 is 5 because 7 – 2 = 5.
• Number lines can be used to find the difference between two numbers.
• Start at the smaller number and count on in jumps until you reach the larger number.
• There are 5 jumps from 2 to 7 and so, the difference between 2 and 7 is 5.

Supporting Lessons

# The Difference Between Numbers

## How to Find the Difference Between Two Numbers

To find the difference between two numbers, subtract the smaller number from the larger number. For example, the difference between 5 and 3 is 2 because 5 – 3 = 2.

No matter how large the numbers are, subtracting the smaller number from the larger number will always calculate the difference.

For example, find the difference between 80 and 20.

The larger number is 80 and the smaller number is 20.

Therefore to calculate the difference between the numbers, subtract 20 from 80.

80 – 20 = 60 and so, the difference between 80 and 20 is 60.

## Teaching How to Find the Difference Between Two Numbers

To teach finding the difference:

1. Use physical objects such as counters to help count on
2. Use number lines to show the size of numbers
3. Show that subtraction gives us the difference

When introducing the idea of a difference between numbers, it is best to start with very small numbers that can be represented with physical objects, such as counters.

Placing the counters side by side can be used to show the difference in size between one number and another.

For example, in the image below we can see that 4 contains 3 more counters than 1 and so, the difference between 4 and 1 must be 3.

After the idea of what the difference between two numbers means is understood, it is time to increase the size of the numbers and look at numbers on a number line.

Number lines are useful tools to use for teaching the difference between two numbers. Mark both numbers on the number line and count the number of jumps needed to move from the smaller number to the larger number.

For example, the difference between 6 and 3 can be seen to be 3 because there are 3 jumps needed to move from the smaller number to the larger number.

## How to Find the Difference Between Two Numbers Formula

The formula for the difference between two numbers, a and b is a-b. Where a is the largest number and b is the smallest number.

For example, the difference between 25 and 13.

## The Difference Between Two-Digit Numbers

To find the difference between two-digit numbers, subtract the smaller number from the larger. With two-digit numbers, the subtraction is most easily calculated using vertical column subtraction.

For example, calculate the difference between 56 and 33.

Line up the digits and subtract vertically.

In the ones column, 6 – 3 = 3.

In the tens column, 5 – 3 = 2.

56 – 33 = 23.

## How to Find the Difference Between Two Odd Numbers

The difference between two odd numbers is always even. For example, 25 – 13 – 12, which is even.

All odd numbers have a difference which is even. This is because consecutive odd numbers always differ by 2. Adding 2 to an odd number takes us to the next odd number.

## How to Find the Difference Between Two Even Numbers

The difference between two even numbers is always even. For example, 16 – 12 = 4, which is even.

All even numbers have a difference which is even. This is because even numbers are all multiples of 2. Every even number is in the two times table. To get from one even number to the next, add two.

## How to Find the Difference Between Two Negative Numbers

The difference between two negative numbers is the same as the difference between these numbers with positive signs. For example the difference between -8 and -1 is 7 because the difference between 8 and 1 is 7.

Now try our lesson on Subtraction Number Sentences where we learn how to write subtraction number sentences.

## 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 ×$3 than work out 5 × $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 × 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 × 5 = 20, 40 × 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 × 4.85 can be estimated as 40 × 5 which equals 200. The correct answer is 189.15. Here is another example of estimating multiplication. Estimate 482 × 734. 482 can be estimated as 500. 734 can be estimated as 700. To calculate 500 × 700, multiply 5 × 7 to get 35 and add on the four zeros found in 500 and 700. 500 × 700 = 350000 and so, the estimate to 482 × 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 ÷ 4 can be estimated as 16 ÷ 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 ÷ 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 ÷ 4. 20 ÷ 4 = 5 and so 200 ÷ 4 = 50. Here is another example of estimating division. Estimate 19 ÷ 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 ÷ 3 = 6 and so, 19 ÷ 3 is just a little larger than 6. 19 ÷ 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 × 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 × 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 × 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 × 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 ÷ 2.03. The numbers can be estimated as 8 ÷ 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 × 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 × 21/4 can be estimated as 5 × 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 ÷ 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 ÷ 21/5 can be estimated as 4 ÷ 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 × 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 × 60 = 180.

The formula to convert hours to minutes is: minutes = hours × 60.

For example, convert 4.5 hours to minutes.

4.5 × 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 ÷ 60 = 2.

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

For example, convert 240 minutes to hours.

240 ÷ 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 ÷ 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 × 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 ÷ 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 × 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 Approximation for Subtraction

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.

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