Lesson video

Hello, one and all, and welcome to another maths lesson with me, Mr. Gratton.

Thank you so much for joining me today.

In today's lesson, we will be looking at different ways of using a pie chart to represent data using proportional reasoning to calculate the angles in a pie chart and using angles that we have calculated to draw a pie chart.

Each part of a pie chart is called a sector.

This is a region of circle created by drawing a line from the centre of a circle to its circumference, called a radius, twice in two different locations.

Imagine each sector of a pie chart like a slice of pizza.

Here are some other keywords that will be referenced throughout the lesson.

Pause here to familiarise yourself with some of these words.

Our first cycle will be an introduction to pie charts where we look at different types of pie charts and demonstrate its relationship to proportional reasoning.

Here's what a pie chart looks like.

What information can you gain from looking at and analysing it? Pause here to give yourself time to analyse as many details as you can.

So here are some of the basics.

Sector A is the largest sector, while sector B is the smallest of the sectors.

Because of that right angle, sector D is a quarter of the total amounts.

Those were some of the bits of information we could gain from looking at the pie chart, but what's the purpose of a pie chart? Well, a pie chart always has a total of 360 degrees inside it.

This fact will be essential to know later on.

The whole pie chart represents the total frequency or the total amount in a dataset.

The whole pie chart represents everything in that dataset.

Each sector of a pie chart represents a subgroup of the data that we're looking at.

And last of all, the size of each sector is proportional to the frequency or value of the group or category that it represents.

The bigger the sector, the higher the frequency or amount.

So in the pie chart on screen, sector A is the largest sector, and so it represents the highest frequency or value of all the subgroups in this dataset.

We will cover two types of pie charts in today's lesson.

The first will have sectors pre-drawn onto the circle, and our job will be to assign data to each of those sectors, whilst the second type of pie chart will be empty to begin with.

Our job with this second type of pie chart comes in two parts.

The first part is to use proportional reasoning to calculate how many degrees the sector representing each subgroup should be, whilst part two is to use a protractor to accurately draw these subgroups on for ourselves.

Because each type of pie chart will still represent the same thing, both of these methods are consistent when saying the larger the sector or the greater the number of sectors, the greater the value or frequency that part of the pie chart represents.

A teacher asked 10 students what their favourite colour was in order to plan the colour scheme of their classroom.

Here are the results.

On this pie chart of 10 sectors, how many of these sectors would represent the number of people who voted for purple? Well, this should be fairly simple, four sectors for four people.

Here's what the rest of the pie chart would look like.

The teacher wants to continue collecting data.

With the teacher's small support class of five students, these were the results.

The teacher tried to represent this information in another 10-sector pie chart.

Can you spot the problem if the teacher uses exactly the same method as before? Ah, here's the problem.

Five students means only five of the sectors of the pie charts are filled in.

This is a problem because a pie chart must be fully filled in.

Having empty sectors defeats the purpose of a full pie chart representing the whole or the entirety of a dataset.

Okay, let's reset that pie chart back to empty.

We can figure out how to fill in this or any pre-sectored pie charts by using a ratio table.

For the purpose of today, the headers for the two columns will always look similar.

In the middle column, we'll always have the words frequency or value, and the rightmost column will have the word sectors or angles when we consider the second type of pie chart later on in the lesson.

Step one is to fill in all the information that you currently know.

In this case, the frequency is five, and there are a total of 10 sectors in the pie chart.

The relationship from left to right is to multiply by two.

We want to consider the purple sector first.

There is one purple data point, and because the relationship is times by two, this will equal two sectors, as you can see on the pie chart.

The same logic applies for green.

Two people voted for green and two times two is four.

So four sectors represent green.

The same logic can apply for pink and blue.

We know for certain this pie chart is complete, as all sectors are filled in with no leftover data from the data the teacher had collected.

With a different class, 20 students voted, as you can see in this tally chart.

What would the pie chart look like if the teacher tried to represent one person's vote with one sector? Well, four people voted purple, so by that logic, four people will be labelled purple.

Next up, six for green.

Oh, I'm already out of space.

The whole pie chart is now filled in, but pink and blue haven't even been considered.

The ratio table approach will work here as well, and it will work anywhere no matter what the frequency of the data is or how many sectors a pie chart has.

Let's fill in this ratio table for 20 students and 10 sectors.

Let's go for pink this time.

Eight students voted for it, and from 20 to 10 there is a multiplicative relationship of times by half, and so half of eight is four.

Therefore, pink requires four sectors.

In real-world applications of pie charts to represent data, it is rare for a dataset to have 10 or 20 data points in it.

A more accurate set of data would look like this with a total frequency of 180.

What frequency would one sector represent? Let's have a look at a ratio table for this example.

A frequency of 180 has to be divided into 12 sectors.

I'm looking to represent one sector, which can be shown like this.

We can use the ratio table in any direction, but the best directions always start from known values and head into the direction of what is unknown.

In this case, we have started on the number 12 and considered that there are 15 12s in 180, and so 1 times 15 equals 15.

One sector represents a frequency of 15.

It is also correct to look at this direction.

12 divided by 12 is one, this time going down rather than left, and so 180 divided by 12 gives you an answer of 15.

Both ways are valid, and both will give you the same answer.

Right, first check, 24 students were asked their favourite colour.

How many sectors of this 12-sector pie chart would represent green? Pause to consider the relationship between all of these numbers.

So the multiplicative relationship is times by a half.

Half of six is three, and so six green votes would represent three green sectors.

Next check, a head teacher collects information from all the students in a school.

The teacher then represents all the data in a pie chart with nine sectors.

Each sector represents 20 of the students from the school.

How many students are in the school altogether? Pause to think of how you could represent this information on the pie chart to the right.

20 students per sector times by the nine sectors in the pie chart, well, that is 20 times 9 equals 180 students in the school in total.

Here's the final check for this cycle.

A teacher asks students whether they wanted the noticeboard to be decorated in their favourite colour.

Here are the results.

A teacher wanted each sector of data to represent four students.

How many sectors are needed in total? Pause to think how you'd represent this information in this ratio table.

Here's the method to fill in that ratio table to get to your answer.

There are 44 students in total in the results.

I know that, in one sector, four students should be represented.

The relationship is therefore times by 11 if I'm going from the bottom row to the top row.

1 times 11 is 11.

Therefore, 11 sectors are needed for the pie chart.

On to the practise, answer Sophia and Lucas's questions about pie charts.

Feel free to draw a ratio table to support your thinking.

Pause now to give yourself time to do this.

Jun collects data and wants to represent it in a pie chart of six sectors.

Pause the video to answer Jun's two questions.

You can draw another ratio table to support your thinking for this question.

Izzy collects data on 150 days.

Each one sector represents six days' worth of data.

Pause to calculate the number of sectors there are in total.

Aisha collects data on 160 days.

In her pie chart, 27 sectors represent yes.

This means the days that had over two hours of sun.

By drawing a ratio table, figure out how many sectors there would be in total.

Pause now to do this.

And last question, Alex wanted to represent 31 days' worth of data in an eight-sector pie chart.

This is impossible.

Can you explain why and explain whether it is possible for Alex to represent this data in any pie chart with equal-sized sectors? And if so, how many sectors must this pie chart have? Pause to explain your answers to these two questions.

For question one, you might have had the calculation 24 divided by 8 equals 3.

3 is the correct answer.

For question number two, you should have had as part of your calculation 12 times 50 equals 600.

600 is your correct answer.

For three A, 30 divided by 6 equals 5 is your first calculation, and for part B, 20 divided by 5 equals 4 is your correct calculation.

For question number four, 150 divided by 6 equals 25 for 25 sectors is your correct calculation.

And for question number five, well, here's a couple of steps.

135 divided by 27 equals 5.

This represents the number of days in each sector.

For the next step, if there are five days per sector, then 160 total days divided by 5 is 32 total sectors.

For question number six, it is impossible because you cannot divide 31 days into eight sectors in a way which each sector represents an integer number of days.

For part B, you can only create a pie chart from Alex's data if the pie chart has 31 sectors or a multiple of 31 sectors, such as 62 or 93.

Our second cycle deals with the second type of pie chart, an empty one where we have to calculate the angle of each sector ourselves.

But why should we do this when we can use pre-sectored pie charts? Well, that's why.

The larger the frequency for the data that we have, the more sectors we risk having to use, and filling in all 40 sectors of that pie chart seems like a lot of effort.

Key pieces of information for this part of the lesson, first, in order to calculate the angles required for each sector, you must, one, find the total frequency or the total value.

If given a table of results, add an extra row to show this total frequency.

In this example, the total value adds all of the distances together, giving a total of 40 metres.

Show this total in its own row.

The second vital piece of information, the total angle in degrees for a pie chart, is always 360 degrees.

Furthermore, by adding an extra column, the angles column, we can easily modify this frequency table to make it a ratio table.

This allows us to demonstrate our proportional thinking far more clearly.

We know that the whole pie chart of 360 degrees represents all 40 pieces of data.

So the most straightforward piece of data that we can fill this table in first of all is 360 represents the total angle.

So we put it at the very bottom of the angle column.

Now, what multiplier links a distance of 40 to an angle of 360? We can do the calculation, 360 divided by 40 equals 9, and so the multiplier is 9.

Filling in the rest of this table is as straightforward as multiplying each of the distances by that exact same multiplier.

It's always important to check that you've done this correctly.

Add up all of the angles except for 360, and the total should always come to 360 degrees.

If it does, you are more likely to be correct in your methods.

However, not every set of data collected will result in a nice multiplier.

Some will result in decimals or fractions.

In this example, we can find the multiplier by dividing 360 by 132.

This fraction, using a calculator if you need to, simplifies to 30 over 11.

I advise leaving the multiplier, as a fraction as introducing decimals this early in our calculations may lead to rounding issues.

As with before, I then take my multiplier, and I multiply each frequency by that multiplier.

If you are using a calculator, make sure you are typing 60 times 30 over 11 correctly with the 60 multiplied by either being to the left of the fraction or as part of its numerator, so 60 times 30.

You should never put this 60 multiplied by as part of the denominator, as this calculation is not the same as the other two that I mentioned.

If your answer is not an integer or whole number answer, make sure to round to the nearest half degree because it is nearly impossible to plot angles more precisely than half a degree using a protractor.

In fact, trying to plot to a whole degree is tricky enough.

So here are the rest of the angles calculated using 30 over 11 as your multiplier.

Ah, notice how diesel has an integer angle.

It is possible that fractional multipliers may still result in integer angles.

Next up, check the accuracy of your angles by ensuring they still sum to 360 degrees.

In this question, yes, they do, so I'm confident the calculations have been correctly applied.

Note, however, it is possible that after all of the rounding, the sum to 360 might still be half a degree out, so 359.

5 or, more likely, 360.

5.

This is normal and just an aftereffect of the rounding that you might have to do.

Protractors are not precise enough that this half a degree will have any meaningful impact on the accuracy of your pie chart.

Check one for this cycle, which of these is the first step when finding any angle in a pie chart question? Pause here to look through all of these important steps.

And the correct answer is B.

Calculating the total frequency is vital in comparing to 360 for proportional reasoning.

On to our next check, what is the multiplier for this ratio table? Pause to consider these options.

Very well done if you got the answer C.

How many 90s are in 360? 360 divided by 90 is the multiplier of 4.

Using the same example, which calculation shows the angle which represents market? Pause to think what the definition of a multiplier is.

Very good spot if you found B to be your answer.

A frequency of 12 times by a multiplier 4 gives you an angle of 48 degrees.

On to the practise, pause the video to give yourself time to fill in each ratio table of values to calculate the angles in a pie chart.

This does include calculating the multiplier below each ratio table.

Here's question three, similar again.

Fill in all the values in the ratio table, including the now non-integer multiplier.

Pause to give yourself time to deal with the fractions and round the decimals carefully.

And question four, the school of Oakfield Academy wants to raise money to improve environmental sustainability in the local area.

A student, Sam, stands by the school gates and asks the parents during morning drop-off whether they would support any fundraising that the school does.

Here are the results.

Calculate the angles needed for this pie chart.

Pause now to begin your proportional thinking.

And on to the answers, for question one, we have angles of 185, 110, and 65 degrees.

For question two, the multiplier is 6, giving angles of 72, 138, and 150 degrees.

For question number three, the multiplier is 9 over 2, or 4.

5.

Even though the multiplier is a fraction, all of the angles still end up as integers with angles of 180, 108, and 72.

For question number four, the multiplier was 9 over 4, or 2.

25.

This gives the angles of 270, 36, and 54 degrees.

So what do we do when we've found all of these angles? Well, we have to construct the pie chart itself.

It will be essential for you to have a protractor to measure the angles and a ruler to draw the straight lines of the radiuses, or radii, of each of the sectors.

So how do we start drawing our pie chart? For each pie chart you need a circle with its centre labelled.

If you're using a compass to draw the circle, the point where the compass needle is pushed onto the paper is that centre.

Step one is to take your ruler, draw a line from the centre going all the way left until it reaches the circumference like so.

Step number two, your protractor needs to be placed precisely in one location.

Overlay the reference point at the bottom centre of the protractor with the centre of the circle like this.

Then align the zero on your protractor with the straight line you just drew.

If your protractor has two zero-degree signs on opposite ends of the protractor, then you must always use the one on the left or the outside of the protractor so that the position of the protractor is upright like in this diagram.

Do not use the right or inside zero for any pie chart that you draw in today's lesson.

Whilst keeping your protractor still, take your first angle, in this case the 120 degrees, and find it starting from the zero degrees on your line, and always read clockwise.

Mark with an X to show the location of that 120 degrees.

Remove your protractor.

Then, with your ruler, draw a straight line from the centre to the circumference through the cross that you just made.

And so this is now one sector of your pie chart completed.

This sector represents purple since purple must have an angle of 120 degrees.

Therefore, label this sector purple.

You must label every sector of your circle when you've drawn it so that you know which sector corresponds to which subgroup of data you have collected.

Now, on to the second sector for green.

Place your protractor back onto the pie chart, placing the reference point onto the circle centre and aligning the zero to the most recent line that you have drawn, the one that closed the purple sector.

Notice your protractor will have a different orientation each time it is used for each of the new sectors.

And next, take the angle for green.

Start at zero, find the angle, and then mark with a cross.

Next up, remove your protractor and draw another line from the centre to the circumference through that cross that you just drew.

Label this semi-circular sector green.

Notice how there is only one sector left for blue.

However, you already have this final sector from the leftover space of the circle not included in the other two sectors for purple and green.

Check the angle of the last sector using your protractor once more.

Place your protractor with the centres aligned and zero on the most recent line that you drew, the line closing the green sector.

Read the angle, which is 60 degrees.

Because the 60 degrees on the protractor matches the 60 degrees in my table of information, this confirms the accuracy of my pie chart.

And so that last sector is a correctly sized blue sector.

Okay, on to our final set of checks, starting with the first step, put these steps into the correct order for constructing one sector of a pie chart.

Pause now to consider all of these steps.

The correct answer is B, then D, then C, then A.

Draw a line going left first.

Then, line up the centre of the protractor with the centre of the circle.

Next up, line up the zero of the protractor with the line that you drew going left.

Then, find the correct angle on the protractor and leave a cross.

Okay, on to the next check, which of these locations most accurately shows the angle 55 degrees? Pause to make your choice.

And the correct answer is B.

Remember to start at the leftmost zero and read clockwise.

So after Izzy plots her first sector, she tries to plot a second.

However, Izzy's method is wrong.

Pause here to have a look and identify why.

So the protractor has not been lined up properly.

Zero needs to rest on that blue line, the line that she has most recently drawn.

On to the last check, Izzy wants to plot 77 degrees with the protractor now placed correctly.

Pause here to identify the correct location of that 77 degrees.

C is the correct answer.

Very well done if you got that correct.

Okay, on to the final set of independent practise questions.

Make sure you have your protractor ready.

The angles have already been calculated and the sectors drawn on.

Use a protractor to figure out which sector links with each key stage.

Pause now to measure the angle of each sector.

So question two, the angles have already been calculated.

Pause now to use a ruler and protractor and correctly draw and label all of the sectors of this pie chart.

On to question three, most of the angles have already been calculated.

Pause here to first complete the missing values of the frequency table and then correctly draw and label all of the sectors of this pie chart.

Question number four, none of the angles have been calculated.

Pause to use proportional reasoning to calculate the size of the angles and draw the full pie chart.

On to the answers, the key stages, the sectors are distributed in this order.

And the pie chart for question two looks like this, with primary having a semi-circular sector and college having a sector greater than one right angle.

For question number three, boots has a sector almost at a semi-circular size, whilst flip-flops is a very small sector.

For question number four, the one that you had to create from scratch, the angles of the pie chart were 96, 156, 72, 12, and 24.

Very well done if you calculated those to begin with.

And the completed pie chart looks like this.

That is superb work following through the range of techniques developed during this lesson, including using proportional reasoning to fill in pre-sectored pie charts and using proportional reasoning to calculate the angles required to draw the sectors of a pie chart from scratch.

And lastly, well done for using a protractor and ruler to physically construct those pie charts.

That is all from me today.

Well done on completing a very technical lesson.

I hope to see you soon for some more exciting maths.

Have a good day.