Lesson video

Hello one and all.

Welcome and thank you for joining me, Mr. Gratton, in this, a Numerical Summaries lesson.

In this lesson we will: check understanding of how to construct and interpret a pie chart.

Pause here to check some of the circle and pie chart based key words that we'll be using today.

First up, let's have a look at how we can find the angles required for a pie chart from a given data set.

Here we have a data set showing the locations of where people live.

Andeep thinks he knows how to draw a pie chart from this information, but he says draw a sector of 65 degrees, 21 degrees and 4 degrees, all to make a full circle.

Do you agree with what Andeep has done? Because I somehow don't think he's correct and Sam seems to agree with me.

Sam says a full circle has 360 degrees and the sums of these three frequencies is 90, not 360 degrees.

The numbers simply do not match, so these frequencies used directly as angles will not make a full circle.

In order to see what angle you need in a pie chart, we need to create a ratio table to find a proportional relationship between the frequencies and the angles.

If given a frequency table, then just draw another column to represent the angles that we want to find out.

Furthermore, it is essential to draw on a totals row since we always know the total angle in a circle or a pie chart is 360 degrees, and we can use this row to great effect later on when identifying these proportional relationships.

In order to find the angle representing each group, we must find the total frequency and place that total in the new totals row and then calculate the multiplier between the total frequency and the total angle of 360 degrees.

The relationship between the total frequency and the total angle can then be applied to each individual frequency to calculate its associated angle.

In this example, the total frequency is 90, therefore the multiplier from 90 to 360 is the fraction 360 divided by 90 or a multiplier of times by 4.

We can then apply this multiplier to each frequency to get its associated angle.

So 65 times by 4 is 260 degrees.

The same applies for the other two frequencies giving 84 degrees and 16 degrees respectively.

A great way to check if our calculations are correct is by adding all three of these angles together.

If the sum of these angles is 360 degrees, then our calculations are likely correct.

Okay, here are a few quick checks.

Pause here to find the values of A and B.

A is always the total of the frequencies above it, in this case, 120, and B is always 360 degrees because the angles in a pie chart always sum to 360.

And pause here to find the multiplier and use it to calculate the angle for coffee.

There are three 120's in 360 and therefore the frequency of 34 times by 3 is the angle of 102 degrees.

And lastly, pause here to consider if all calculations are done correctly, what will the sum of these four angles be? The sum of all of the angles in a pie chart is always 360 degrees.

This is a good way of checking that all of your calculations are correct.

If given a description of values, then we can create a frequency ratio table to help support your calculations.

So for this survey, whose responses are either agree, disagree, or unsure, we can create this table whose first column includes the responses or names, whatever data you are collecting.

We can also create a frequencies column and an angles column.

And don't forget the totals row.

The only information that we need to construct and complete a full table are the given frequencies and the knowledge that 360 degrees is the total angle inside a pie chart.

Sometimes the multiplier between the total frequency and the total angle 360 degrees is a fraction or a decimal.

Don't bother rounding any angles that you calculate from this to any decimal places, round to the nearest integer.

It's hard to measure decimal angles when using a protractor like this.

So for our example, our total frequency is 94, therefore the multiplier is 360 over 94.

Whilst this fraction does simplify to 180 over 47, it does not simplify into an integer.

So when doing our calculations, especially when using a calculator, you can just use 360 over 94 rather than any simplified version, there is no difference.

Multiplying each frequency by its multiplier gives an integer of 180 degrees.

It is still possible to get an in integer answer even with a fractional multiplier.

We then get this decimal that rounds to 69 degrees and the remaining decimal that rounds to 111 degrees.

In the vast majority of cases, the sum of the rounded angles still equals our 360 degree total.

For this check, pause here to consider what are the values of A and B.

A is the total 63 and B is the fraction 360 over 63, or any other simplified version such as 40 over 7.

However, keeping this as 360 over 63 is absolutely fine when using it as a multiplier to calculate some angles.

Okay, next up, pause here to consider the values of C and D rounded suitably.

Here are the answers.

Remember, round these angles to the nearest integer.

Great stuff, onto some independent practise.

Pause here to find the angles for all data in each of these four data sets.

As each question progresses, you'll need to add more information to each table and potentially draw more columns and rows to your table.

Great effort on those four questions.

Pause here to compare all of your calculations and answers to the ones on screen.

Now that we know the angles that we need to construct a pie chart, let's have a look at actually constructing one.

So get those protractors at the ready.

Andeep once again jumps right in and maybe misses a few steps to constructing a pie chart properly.

Andeep places the protractor onto the circle.

Notice how he does so correctly by overlaying that middle part of the protractor with the circle's centre.

Then he marks on a 52 degree angle, a 128 degree angle and a 180 degree angle and draws three lines from the centre to those three angles to get this pie chart.

Pause here to think about or discuss what Andeep has done wrong here.

We'll explain the correct method fully in just a minute.

But first of all, let's take a closer look at the pie chart that Andeep drew.

Do any of these angles look remotely close to a 128 degree angle? Definitely not.

None of the angles in this pie chart are obtuse, so we know for certain that something has gone wrong here.

Let's construct this pie chart together.

If you need to, pause the video now to find or construct a circle.

An important first step that is easily forgotten is to draw a reference line segment.

This line segment goes from the centre of the circle to its circumference.

I personally like to draw this going vertically upwards, but some others like to draw it going to the right from the centre instead.

Any direction is absolutely fine, but it's best to keep it consistent across all the circles that you draw to get into a good routine when using your protractor.

After drawing that reference line, do the one thing that Andeep actually did correctly and place your protractor so that the centre bottom of the protractor, usually shown by that symbol or a set of cross hairs, and make sure that it aligns exactly with the centre of your circle.

If a circle is already given to you, the centre should always be marked.

If you are drawing a circle for yourself with a pair of compasses, then the point that the compass needle digs into the paper is the circle's centre.

Making sure to keep that point on the protractor overlapping with the centre of the circle, Rotate the protractor so that one of the 0 degree measurements lines up with your reference line like so.

Notice how this 0 degree line lines up perfectly with the reference line on the circle.

Again, consistency is essential.

Right now, look whether you've lined up with the outer or inner 0 degree measurement.

This might be different depending on the exact protractor that you are using.

For example, I am using the outer 0 degrees.

I must then continue to use the same outer 0 degrees throughout this whole pie chart.

Otherwise, I risk drawing an angle in the wrong direction and overlapping a previously drawn angle.

Now that we have properly aligned our protractor, choose an angle from your ratio table and, starting at the same 0 degrees that you identified earlier, find that angle.

In this example, I will start at the outer 0 degrees and find the 52 degree angle.

I then mark this location with a pencil.

It's okay if the location that you mark is outside the circle itself or inside the circle.

The location doesn't matter as long as it is at the right angle.

Pause here to try this yourself for your circle.

Then from the centre of your circle, draw a line segment to the point that you marked.

The angle you just made is 52 degrees.

Pause here to draw this line segment and label the angle.

Here's one of the most important points that you need to be very careful about, the thing that Andeep got wrong earlier.

When starting to draw a new angle, a second angle, in this case, make sure you orientate your protractor so that the same 0 degrees aligns with the most recent new line that you drew, not an older one.

So for this pie chart, I should not align the protractor with the original reference line.

Rather, I should align my 0 degrees with this new purple line segment that we drew to complete our 52 degree angle.

So from there it is the same again.

Starting at your chosen 0 degrees, mark the next angle on your pie chart.

For us, it is this 128 degree angle, which I will mark here.

As with before, draw your line segment from the centre of the circle to this marked location.

This second angle is a 128 degree angle.

Pause here to draw this line segment and label the angle.

Notice how there is no need to construct the final angle in any pie chart.

The remaining angle should be the exact same angle that you want to construct.

You don't need to construct the angle, but it is very good practise to check that the remaining angle is the size that you expect it to be.

As with before, do not line up your protractor with an older line segment.

Rather, line it up with the most recent line segment that you drew, in this case, the dashed blue line.

We can see that this 180 degrees on the protractor matches up with the 180 degrees in the ratio table.

Notice how this final angle aligns with the original reference line that you drew at the beginning of this pie chart.

After checking that all the angles are correct in size, label each angle with its associated name from the data set.

If you've been able to follow all the steps correctly, then the order in which you constructed your angles should match the names of the three meals in the same order.

So breakfast, then lunch, then dinner.

Make sure to use some angle sense, just to check though.

Breakfast is an acute angle, lunch is obtuse and dinner is the largest angle at a 180 degree straight line.

Okay, let's see if you could apply these instructions in order.

Starting with the first instruction, pause here to put these instruction steps in order.

And the correct order is BDAC.

Pause here if you need to make a note of this.

Izzy wants to plot an angle of 55 degrees.

Pause here to choose the letter that correctly represents the point that she should mark.

Remember to always start at 0 degrees that is aligned to the current line that you are using to create your angle.

So in Izzy's case, she's using that outer 0 degrees and so the outer 55 degrees is what she needs to use to create her 55 degree angle, which is the point at B.

Izzy has correctly drawn her 55 degree angle.

Izzy has now marked the point that will create her second sector with an angle of 34 degrees.

However, Izzy has done this second step incorrectly.

Pause here to think about or discuss what she has done wrong in this diagram.

Izzy has not rotated her protractor around so that the 0 degrees lines up with the most recent line that she drew, rather, she's kept it in the same place, meaning currently it is in the wrong orientation for her to draw a new angle.

The current orientation of her protractor means that this 34 degree angle will overlap with the old 55 degree angle that she drew previously.

And lastly, pause here to identify which of these locations should be marked to create a segment with a 24 degree angle.

And the answer is A.

If you have a protractor whose maximum angle is 180 degrees, then to construct an angle of greater than 180 degrees, you can split the angle up into 180 degrees plus the remaining angle.

So for this 269 degree angle, we can split it up into 180 degrees and the remaining 89 degrees.

To then draw this reflex angle, align your protractor with the most recent line segment that you drew, just like before.

Then mark a 180 degree point, but don't draw a line segment.

If it helps for you to do so, then draw a faint line from the centre to that marked 180 degree point.

Then all you need to do is realign your protractor to this new point and mark on the remaining angle, in this case 89 degrees.

The reflex angle is the correct angle, in this case, 269 degrees.

For this quick check, which two angles can I split this 188 degree angle into in order to make this angle possible using this here protractor? I can split it up into 180 plus 8 degrees.

Great stuff, time for you to practise constructing some pie charts.

Pause here to complete the ratio table, then construct a pie chart from the angles that you found.

And pause here to do the same for question two, but this ratio table has more for you to fill in.

And for question three, construct two pie charts, then use the pie charts to explain how you know that the sample is not proportional to the population.

Pause here to do this.

Great effort on constructing those pie charts.

Pause here to check your answers to question one and see if your pie chart has similar looking angles to this one.

And pause again here to compare the ratio table and pie chart for question two.

And finally question three, pause here to have a look at all the information on screen and note that the sample is not proportional to the population because the two pie charts are not similar to each other.

That is to say the same group does not have the same angle in both pie charts.

Now we know how to construct a pie chart.

What if a pie chart was given to us instead, what information can we take from it? A multiplier shows the relationship between the frequencies and the angles and can be used in both directions, either to calculate an angle from a frequency or a frequency from an angle.

For example, a frequency of 9 times by a multiplier of 6 gives an angle of 54 degrees.

This same multiplier of 6 goes from a frequency of 24 to an angle of 144 degrees.

For this data set, from any frequency to its angle, the multiplier is always a times by 6, meaning that 12 times by 6 gives you the angle of 72 degrees.

However, the multiplier in the opposite direction is a times by a sixth or a divide by 6.

This allows us to find the frequency of pink by dividing the angle of 90 degrees by 6 to give us a frequency of 15.

However, you must already know either one frequency or the total frequency in order to find all the other frequencies from their angles.

It is impossible to calculate frequencies from the angles in a pie chart if none of the frequencies are known.

For this check, pause here to consider what calculation finds the multiplier for this data set.

The only datum that I know both the frequency and the angle for is the dry datum.

Therefore, the relationship from frequency to angle is a times by 5 or 60 divided by 12 is 5.

Now using the multiplier, pause here to consider what the frequency of days with heavy rain is.

If frequency to angles is a times by 5, then angles to frequency is a times by a fifth or a divide by 5.

Therefore, the frequency of heavy rain is 90 divided by 5, which is 18.

And now, pause here to find the values of A, B, and C.

The number of days with light showers is 16.

Let's try this exact same question again, but in a different way.

Pause here to find the values of D and E and check that the number of days with light showers is still 16.

It is.

I can find that same missing frequency in two different ways and still get the same answer.

Here are the results from an investigation of two schools represented as pie charts.

Pause here to think about or discuss in which school did more pupils walk to school.

Ah, this is a bit of a trick question.

Pie charts show proportion.

For example, what percentage of students walk to school.

However, it tells absolutely nothing about frequency.

For example, how many students walk to school.

The total frequency of one pie chart could be significantly larger or smaller than the other pie chart, like so.

Rowanwood has a much, much higher number of pupils than Oakfield, meaning that whilst a much lower proportion of students at Rowanwood walked to school, there are still more numbers of students who walked to school at Rowanwood than Oakfield.

In order to compare two or more pie charts, we need to measure the angles in each pie chart and know at least one frequency or total frequency from each pie chart.

Let's use a protractor to calculate the angle of each sector in both pie charts.

So for Oakfield, we have these two angles, and for Rowanwood we have these two angles.

We have successfully measured the angles in each pie chart.

We can then represent all known information in a ratio table.

As we start to get given more information, we can input it into the correct ratio table.

So for Oakfield, 600 were away supporters, the 600 goes here.

The 4,800 total people for Rowanwood goes here.

We can find all other frequencies from both data sets because along with each pie chart, one frequency was given.

Let's see which team had more home supporters.

The multiplier for Oakfield to get from angle to frequency is 600 over 72.

You can simplify this, but you don't need to, especially when using a calculator.

We multiply the home angle of 288 degrees by this multiplier to get 2,400 home supporters.

The same can be said for Rowanwood's multiplier of 4,800 over 360 to give 2,000 home supporters.

Looking at the home supporters now, Oakfield had 400 more home supporters.

Pause here to choose true or false and the correct justification for your answer to this statement.

The answer is false, because two identical pie charts can still represent different frequencies because each pie chart may represent a different population.

Amazing, onto the final few practise questions.

For question one, use a protractor in order to complete each ratio table and calculate all frequencies in each data set.

Pause now to do this.

And finally, question two.

The minor or smallest sector of each pie chart represents 120 people.

Pause here to consider which sample had a larger sample size and by how many people.

Great effort on analysing information from these pie charts.

For question one, pause here to check your answers and compare them to the ones on screen.

And for question two, sample A had 80 more people than sample B.

Pause now to compare your ratio tables of information to these two on screen.

Amazing work everyone on the diverse amount of pie chart related content you've covered today, such as comparing total frequencies of a data set to 360 degrees, the total angle of a pie chart.

And using proportional reasoning to find the angle required for each value of data from its frequency.

We've also seen how to construct a pie chart from scratch using a protractor.

We can also use angles in a pie chart to find its associated frequencies, assuming at least one frequency is already known.

And that two pie charts with sectors that are exactly the same size may still represent a totally different frequency.

Thank you all so much for your effort today.

I'll see you again soon for some more maths, but until then, have an amazing time.

Take care and goodbye.