Lesson video

Hello everyone and welcome to another math lesson with me, Mr Gratton.

In today's lesson we will be interpreting pie charts by looking more deeply at ratio tables that represent pie charts.

As well as looking at what we can and cannot summarise when making comparisons between two pie charts.

Here are a few words that will be very helpful for you to know for this lesson.

Pause now to familiarise yourself with some of these words.

To begin with, we will calculate the frequencies that a pie chart represents either based off of a ratio table or by measuring the angles in a pie chart using a protractor.

But first, let's look at a quick recap of how we can use ratio tables to create a pie chart to begin with.

The multiplier is a number that you can multiply a frequency by to get its corresponding angle.

This multiplier is the same for every frequency in a table.

In this example, the multiplier is 6 because each frequency can be multiplied by 6 to get its corresponding angle.

For example, 12 times 6 is 72.

The number 360 is both the sum of all of the angles as well as the total frequency multiplied by the multiplier.

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

Onto another ratio table, except there's a lot missing.

There are no frequencies shown at all.

Is it possible to calculate anything from this set of angles only? If you don't know any of the total frequency or any other frequency, then it is impossible to calculate anything new.

This is because a pie chart only shows proportion and not the number of data points that went into creating the pie charts to begin with.

Oh, but the moment you know any one frequency, and this does include the total frequency, you can actually calculate the rest of the frequency column.

Right, let's have a look at how we can do this.

Step one is to calculate the multiplier.

You can do this by focusing on the wales row.

Since this is the only row with both the frequency and the angles known.

You can multiply the frequency by its multiplier to get the angle.

Therefore, you can divide the angle by the frequency to get the multiplier.

An angle of 60 divided by a frequency of 15 equals a multiplier of 4.

Step two, in order to find a frequency, you have to divide by the multiplier as we are finding unknown values in the opposite direction to the one the original multiplier is representing.

176 divided by the multiplier 4 equals a frequency of 44.

And here's the rest, such as 20 divided by the multiplier 4 equals 5.

Remember to always check your work.

Find the total frequency by adding together everything in the frequencies column.

The answer to this, the total frequency should always be equal to 360 divided by the multiplier.

In this question the total frequency is 90 by adding together 44, 5, 26, and 15.

Let's have a look at this second method.

360 divided by 4 because the multiplier is 4.

360 divided by 4 is also 90.

Therefore, I'm confident that the values in this ratio table are correct.

Right, let's check your understanding.

For this ratio table, which calculation finds the multiplier? Pause to look through these options.

And C is the correct answer.

Remember to focus on a row that has both the frequency and the angle labelled.

In this case, dry is the only such row.

5 times 12 is 60, therefore 12 is the multiplier.

Now using this multiplier of 12 in how many days did it rain? Pause to consider what you would do with the multiplier now that you know it.

If going from a frequency to an angle is times by 12, then going from an angle to a frequency is a divide by 12.

If rain had an angle of 120 degrees, then 120 divided by the multiplier, in this case 120 divided by 12, that would equal 10.

So 10 is the frequency, the number of days that it rained.

And final check for this set of data.

I have calculated the four frequencies correctly.

Which two calculations when used together will help you check all of the values in the ratio table are correct.

Pause to look through all of those options and see which two options make sense for this ratio table.

And the answers are A and C.

For A, we are adding all of the frequencies together to get to the total frequency.

For C, we are dividing 360 by the multiplier, in this case 12.

Both A and C get to 30 and therefore we know that all of the values in the ratio table are likely correct.

Great.

This is a lesson on pie charts and whilst we've dealt with its angles, we haven't actually looked at any pie chart yet.

If we do not have a ratio table, we can always create and fill one in for ourselves.

In this example we have a pie chart and one frequency, the total frequency that this pie chart represents.

We can place this total frequency in the total frequency position of our ratio table.

Also, we can always place 360 in this position as the total angle of any pie chart is always 360 degrees.

If we only had the ratio table, we could do very little with it.

But because we have a pie chart as well, we can use the protractor to measure all of the angles and fill the ratio table in with all of those measurements.

Make sure to correctly position your protractor.

We can do this by overlaying the bottom centre of your protractor with the centre of the circle.

And also the zero degrees with one of the lines or radii on the pie chart.

Starting at zero and reading the angles clockwise.

At what angle does another radius appear first? In this case, it is the 75 degrees on the protractor.

Since this angle was measured on the in-town sector of the pie charts, I can place 75 degrees in the in-town angles position of my ratio table.

Remember to change the orientation of your protractor so that the zero degrees lines up with a different radius so that you end up measuring a different angle.

Do not rely on putting your protractor in the same place in the same orientation each time.

You will end up measuring either the same angle again and again or miss measuring a different angle.

In this example, I'm now gonna place my protractor here so that I can measure the out-of-town sector next.

This sector has an angle of 125 degrees so I can fill the appropriate spot in the ratio table two.

Lastly, let's rotate the protractor again so it lines up with the online sector of the pie charts.

Look at the degrees, it measures 160 and so I fill in the ratio table online section with 160 degrees.

When you have found all the angles, the process is exactly the same as before.

Firstly, calculate the multiplier by looking at a row with both the frequency and angle given.

In this example we have 72 and 360 degrees both mentioned in the total row.

So the question is how many 72s are in 360? By dividing 360 by 72 we get a multiplier of 5.

This also means a divide by 5 multiplier going in the opposite direction.

Step two, use the multiplier to divide each angle to convert it into its corresponding frequency.

75 divided by a multiplier 5 gives you a frequency of 15.

And the rest can be calculated in exactly the same way.

Step three, check your answer using one of the checking techniques that we covered before.

In this case, let's add together all of the frequencies to get a value 72.

Where have we seen the number 72 before? Ah yes, it is the total frequency that was mentioned in the original question.

Since my frequencies match the original total frequency, we can assume all of the values are correct.

Onto our next check, which angle is this protractor measuring for strong breeze? Pause here to read the numbers on the protractor.

The answer is 120 degrees.

By looking at the appropriate row, calculate the multiplier for this same ratio table.

Pause to choose the correct information to calculate the multiplier.

The strong breeze row is the only row with both a frequency and an angle.

Therefore the multiplier is how many 15s in 120 or 120 divided by 15.

Regardless, the multiplier is 8.

Which also means there is a divide by 8 multiplier going in the opposite direction.

Last up, using all of this information and an understanding of totals, calculate the missing angle for gale.

Pause to think how you could do this with the given information.

All the angles must add up to 360, so by doing 360, take away 120, takeaway 184, takeaway 48, you get a missing value of 8.

8 is the missing angle for gale.

Right, onto some independent practise.

A group of students were asked about their favourite subject.

25 students voted for science.

Pause now to fill in the given information into that ratio table.

Then calculate the multiplier and after that, calculate all other frequencies.

For question number two, by first calculating the multiplier, find all of the missing frequencies.

Pause now to do this.

Onto question three and you'll need a protractor for this question.

Measure the missing angles for Monday, Tuesday and Friday.

And then by calculating the multiplier fill in all of the missing frequencies.

Pause now to grab a protractor and measure everything that you need.

Onto question number four for some further practise.

Fill in all of the missing information on each ratio table and on each pie chart.

Pause now to consider all of the information for each ratio table pie chart pair.

Onto the answers.

For question one the multiplier was 4 and the missing frequencies were 38, 15, 25, 12 with a total frequency of 90.

For question number two, the multiplier was 3 and the missing frequencies were 21, 50, 10 and 120.

For question three, the missing angles that you have used a protractor to measure were 15 degrees for Monday, 105 degrees for Tuesday and 30 degrees for Friday.

Using the Monday row, since we know both the frequency and the angle, we could calculate the multiplier to be times by 3.

Therefore, by dividing each of the angles by 3, you would get the corresponding frequency.

The frequency for Tuesday is 35, 52 for Wednesday, 18 for Thursday and 10 for Friday.

Here are all of the answers for question four.

Pause the video here to compare all of the numbers on screen to all of the numbers that you had calculated for this question.

Very well done for all of these challenging calculations that you've done with pie charts, but what if we had two pie charts? Could we compare them at all? Well, let's have a look if we can.

Two schools were asked how many of their students walked to school.

Here are the results shown into pie charts.

In which school did more students walk to school? Pause here to think about it by looking at both these pie charts.

Ah, it's actually a bit of a trick question because pie charts show proportion but tell nothing about frequency.

Just because a sector is bigger, it does not mean that the sector represents a higher frequency or value.

But rather it represents a greater percentage of data for the dataset it is linked to.

In this case, a greater percentage of students walked in school A than in school B, and here's the logic in action.

In school A of 200 students, 150 walked whilst in school B of 600 students are 200 students walked.

Whilst 200 is a bigger frequency, it represents a smaller percentage or proportion of the whole school and hence in school B, the walk sector of the pie chart is smaller.

We can summarise all of this information with these sentences.

Whilst a greater proportion or percentage of students in school A walked compared to school B, more students in school B walked because school B had significantly more students.

The largest size of school B was a more significant factor than the lower proportion or percentage of students who walked.

In a second I will show you two pie charts each representing a pizza, the size of each pie chart is proportional to the size of the pizza.

This means that the larger the pie charts, the larger the pizza.

Sam says, if I eat the Hawaiian half of pizza A, I will eat just as much pizza as if I ate the Hawaiian half of pizza B.

This is because I'd still be eating half of whichever pizza.

Sam is wrong.

Can you explain why.

This is because pizza B is smaller and Sam would end up eating a smaller amount of pizza even if they eat half of both.

Eating half of a small pizza is most certainly not the same as eating half of a massive family sized pizza.

Okay, quick check.

Is this statement true or false? If two pie charts both have a sector with the same angle, then they always represent the same frequency.

Pause to make your decision.

The answer is false And now pause to decide which of these explanations justifies it being false.

Two pie charts could be representing two vastly different size of dataset.

Now onto some examples that combine angle problems with comparing between two pie charts.

Two more schools were asked how many of their students walked to school.

Here are the results represented as pie charts with the angles of the sectors written on for you.

Whilst we can't figure out any frequencies yet, we can figure out in which school more students walked if further information was given.

Such as this.

In school C, 100 students did not walk to school whilst in school D, there are a total of 380 students that go to that school.

The question is in which school did most students walk? For school C we can place this information into a ratio table.

For each of these ratio tables make sure to construct the ratio table in such a way that you can put each set of the pie chart and the total of the school into the table itself.

For example, here we have don't walk, walk and total as the three headers with the angles from the pie chart clearly written on.

It also says 100 students did not walk, so 100 goes in the don't walk column.

Proportional reasoning works in both horizontal and vertical directions.

So use whichever direction makes most sense with the numbers given and gives you the easiest numbers to calculate with.

In this case, we can notice that the size of the walk angle is double the size of the don't walk angle.

The same logic can therefore be applied to the frequencies.

So double 100 is 200.

So 200 students at that school C walked.

Now let's look at school D, which has a total of 380 students.

The ratio table will look like this, with the same don't walk, walk and total headers.

We notice that the same number of students walked and didn't walk.

This is the same as saying half of the students walked.

Half of 380 is 190.

That means 10 more students from school C walked than in school D.

200 versus 190.

It is also sometimes possible to identify pie charts using understanding of the real world.

Although you must be careful of the assumptions that you make.

For example, students were placed into two groups during an exam.

Groups of those that revised and groups of those that did not revise.

Which of these pie charts matches with which group? Pause here to consider or discuss your thoughts.

It is more likely that pie chart A represents those that revised as hopefully you are more likely to pass an exam if you revise for it.

Let's use these pie charts to consider the total number of students in each group.

The same number of students from each group failed.

In which group were there more students in total? How do we do this? Well, we can do this by assuming 10 students from each group failed.

I could have picked any number.

It would not change how I perceive and do this question.

I chose 10 as it is an easy number to deal with and a sensible size for the subset of a class.

In group A, if 10 failed, then the number of people who passed must be greater than 10 as the sector representing passed is larger.

Let's call this size 15.

Again, it does not matter which number that I pick as long as it is greater than 10.

This means group A has a total of 25 students.

Then in group B, if 10 failed, then the number of people who passed must also be less than 10 as the sector representing passed is smaller.

Let's call this sector a size of 8.

Again, it does not matter which number I pick as long as it is a number smaller than 10.

This means that group B had a total of 18 students.

No matter which numbers I picked, I will always come to the same conclusion, there were more people in group A.

Here's a check for understanding with some brief setup first.

Students who walked to school and students who didn't walk to school were asked at the school gates whether they lived 5 kilometres away from school or not.

Match the descriptions C and D to the correct pie chart A and B.

Pause now to think of the real world logic behind these pie charts.

Pie chart A matches with D.

If you live further away from school, then you are more likely to not walk to get there.

Whilst in pie chart B, if you live closer to school, you are more likely to walk that shorter distance.

More analysis of these same pie charts.

Sector X and sector Y on the two pie charts represent the same number of students.

Which pie chart represents a larger number of students? Pause to consider the proportional size of sector X when compared to its pie charts and think the same for sector Y.

And also think about which pie chart therefore represents a greater number of students.

As with before, let's assume both sector X and Y represent 10 students.

The 10 students in this pie chart represent a larger proportion of the total pie chart.

Meaning there's less remaining pie chart here for the other students to represent.

Whilst in this pie chart there's more remaining students.

There's a greater proportion left over and so this pie chart, the walked to school pie chart represents the larger total frequency 'cause 10 students represents a smaller part of the total proportion.

Okay, last few independent practise questions.

For question one, the head teacher says, "More students passed the exam in Class R" The head teacher may be incorrect.

Can you explain why? Pause to consider your explanation.

For question two.

In both classes, 15 students passed the exam.

Complete the ratio tables underneath by using the information in the pie chart to work out which class had more students.

Pause to fill in all of this information.

For question three.

People in Leeds asked whether they support Leeds United or not.

They were also asked whether they would watch Leeds United's next football match.

Two pie charts were drawn.

One to show the supporters and one to show the non-supporters.

Which of the two pie charts represents the group of people who supports Leeds United? Explain how you know and pause to look at these pie charts and come up with your explanation.

For question number four.

The minor or smaller section of each pie chart represents 80 people with which group, A or B were more people interviewed? Use the information in each pie chart to complete the ratio tables to support your answer.

Pause here to interpret, transfer, and calculate with all of this information.

Onto the answers.

Whilst for question one, a higher proportion of students passed in class R than Class P, class R could also have a lot fewer students.

This means that 3/4 of a small class could be smaller than half of a big class.

Here are some numbers to demonstrate this.

Class R has a class size of 12 and class P has a class size of 30.

3/4 of 12 is 8.

Whilst half of 30 is 15.

Therefore, more students in class P passed than in class R.

Onto question two.

Class P had 10 more students.

Pause here to compare all of your numbers in your ratio table to those on screen.

Question number three.

B is more likely to represent those that do support Leeds.

As those that do support Leeds are more likely to watch them play a match than those who aren't supporters.

Finally, question number four.

240 more people were interviewed in group B.

In fact, group B had double the number of people in it than in group A.

Pause here to compare the values in this ratio table to those in your own.

That is some great work on some challenging questions and contexts in a lesson where we used the multiplier as both a multiply and dividing calculation to find missing angles and frequencies.

We've also used a protractor to aid in the calculation of frequencies by finding the appropriate angles.

We have also compared pie charts on both mathematical and real world levels to consider totals where possible and identify the source of data of that pie chart.

I appreciate all of the hard work that you've done during this lesson.

I hope you have a great day and I'll see you soon for some more maths.