Exploring the difference between averages

Two tasks designed to get pupils thinking about the different averages and how they are affected by new numbers added to a set. It preempts this lesson on which average you should use and why. Its also quite a nice bit of cross-curricular thinking.

Task 1 - Averages and probability (editable files here)

Task 2 - Match the averages to the description

Probing questions about finding the mean from grouped data

Estimating the mean from grouped data is one of those topics that all my students can do in lesson, but tend to forget in tests. Because it is quite an abstract process, students often learn the steps of the method without really understanding what is happening.

To improve this, I have developed two resources:

• The first can be found here and it enables students to learn how to find the mean from a table without you saying a word! It is very differentiated and I have used it successfully with students at KS3 and KS4.
• The second is the slide below. The idea is to get students to think about how the steps they are calculating relate to their final answer.

The False-Positive Paradox as a Class Activity/Discussion Point

There is a test that screens for some rare disease (only 1 in every million people is infected). It is 99% accurate (it gives the correct answer of positive or negative 99% of the time). You go for the test and to your dismay it returns a positive result!
What is the chance that you actually have the disease?

You may think that its pretty likely you have the disease. However, from the information above, the actual probability that you have the disease is still very unlikely (1 in 10'102). This is the basic premise of the false-positive paradox.

I don't really like that it is named as a paradox, because it is not actually in any way paradoxical, just counter-intuitive. It works because there are so many more non-infected people (999'999) that each have a 1% chance of getting a false positive than the number of infected people (1) who has a 99% chance of getting a true positive result.

This counter-intuition can cause exactly the sort of cognitive-disonance that can be so useful in getting people to realise that their current understanding needs improving, but in its current form its pretty wordy, inaccessible and abstract. I wanted to make it in to a class activity. Instructions below.

 

 

 

How to run the Class Activity/Discussion Point:

• As students enter the room, give them each a slip of paper with a random word on it. Tell them not to show anyone else.
• Once they're settled, tell them that there is a (make-believe) disease going round school that only affects about one student per class. In this class the one student, the one with the word banana on their slip of paper, is infected (Don't tell anyone if its you or not!).
• Tell them you have a way of checking if anyone has the disease, but its not always accurate. To perform the test, you will ask the student if they are infected. Before answering, the student will roll a dice and will tell the truth if they roll a 1, 2, 3, 4 or 5, but they will lie if they roll a 6.
• Perform this test on some students until someone claims to be infected.
• Ask the class how likely is it that this student is actually infected. Ask for justifications. Most will say very likely, some may evan say 5/6.
• Perform the test on a few more students until you get a few more claiming to be infected.
• But they can't all be 'very likely'! What is going on here? 

By the way, make sure the students know the disease is not real!

Would this work in your classroom? Could it be improved? At the moment its just an idea and I'd like to know what you think.

Lesson Sketch: Who Wrote This?

I'm excited about this lesson idea! So excited that it had me tweeting J.K. Rowling and Stephen Fry (though I didn't get a reply yet from either). I hope you like it as much as my pupils did last week.

Act 1

Get pupils to read this letter:

Dear ___________,

I'm so sorry to hear that life is getting you down at the moment. Goodness knows, it can be so tough when nothing seems to fit and little seems to be fulfilling. I'm not sure there's any specific advice I can give that will help bring life back its savour. Although they mean well, it's sometimes quite galling to be reminded how much people love you when you don't love yourself that much. I've found that it's of some help to think of one's moods and feelings about the world as being similar to weather:
Here are some obvious things about the weather:

It's real.
You can't change it by wishing it away.
If it's dark and rainy it really is dark and rainy and you can't alter it.
It might be dark and rainy for two weeks in a row.

BUT

It will be sunny one day.
It isn't under one's control as to when the sun comes out, but come out it will.
One day.

It really is the same with one's moods, I think. The wrong approach is to believe that they are illusions. They are real. Depression, anxiety, listlessness - these are as real as the weather - AND EQUALLY NOT UNDER ONE'S CONTROL. Not one's fault.

BUT

They will pass: they really will.
In the same way that one has to accept the weather, so one has to accept how one feels about life sometimes. "Today's a crap day," is a perfectly realistic approach. It's all about finding a kind of mental umbrella. "Hey-ho, it's raining inside: it isn't my fault and there's nothing I can do about it, but sit it out. But the sun may well come out tomorrow and when it does, I shall take full advantage."
I don't know if any of that is of any use: it may not seem it, and if so, I'm sorry. I just thought I'd drop you a line to wish you well in your search to find a little more pleasure and purpose in life.

Very best wishes, ____________

Tell them that it was written by one of the following people:

J.K. Rowling	Lewis Carroll	Stephen Fry
Russell Brand	Franz Kafka	Fyodor Dostoevsky

Let them discuss who they think it might be and why. Ask if there are any we could discount straight away. (note: I feel that I should have more women up there. Can you think of any women authors that all the kids would know?)
They will talk about writing styles and uses of certain words and things they know about the lives of these authors (I've purposefully used people for whom it would be understandable that they would have written such a letter).

Act 2

We are going to try to use their writing styles to determine who wrote it. What sort of things could we measure about their writing styles? (Some) Possible answers:

• How often they use certain words/punctuation marks (make sure the words chosen are common enough to be in other writings and not specific to the context).
• Average sentence length.
• Average paragraph length.

We will also need some samples to measure these things on. I'd use the previews you can get on Google Books:

• The Beadle and the Bard by J.K. Rowling
• The Life and Letters of Lewis Carroll
• The Fry Chronicles by Stephen Fry
• My Booky Wooky by Russell Brand
• Letter to my Father by Franz Kafka
• Notes from the Underground by Fyodor Dostoevsky

This task works best if each group measures and compares a different word or aspect of the writing as it creates for some great discussions about which measurements hold more weight and why.

Act 3

Write a table of the techniques each group used and the conclusion they made from it. Decide as a group who we think it is. By this point my students were desperate to know who wrote it. Talking points:

• Accuracy and how we could have obtained better estimates.
• Which techniques should hold most weight? Why?
• Why is "The Beadle and the Bard" probably not as useful to use as "The Fry Chronicles"?
• Where might this technique be used outside the classroom?

Reveal the answer: Tadaa!
After this you might also want to show how historians have used this exact same technique (and its name: Stylometry)!

Final Notes: When I used this in class, with student written essays rather than a famous letter, it worked exceptionally well. As I said, pupils were desperate to know who wrote the essay by the end. It fell down a little at the end as I didn't have a good way to show the answer. Making it about celebrities (I think) makes the answer more interesting.
The reason I tweeted Fry and Rowling was to try to get a school essay from them (the story would be that I have an old school essay from a celebrity and I could sell it if only I could remember which of these people had written it). This letter works well and is a great and interesting bit of writing in itself. The only problem I have with it is that students could quite easily Google part of the text and get the answer.

What about you? How could you improve it?

Lesson Sketch: A Deviation from Average Top Trumps *UPDATED*

Firstly, I have to admit that I haven't tried this one out yet. I just got the idea this weekend as I unearthed some 'fractions and decimals top trumps' I made for this Year 7 Maths Attack! I like the top trumps idea for a way to add a game element to practising maths skills (especially these on missing angles), but I also have two problems with them:

• To have enough for a whole class, you need a whole lot of printing, laminating and cutting. Either you do it or it eats in to precious class time.
• There's a low-limit to how hard you can make the problems before the time taken working out if you have won sucks all the fun out of playing.

Enter 'The Modal Group'; a set of Super-Average Super-Heroes!

Before you start, let the class decide on the four stats that you will be comparing (e.g. Strength, Wits, Speed, Etc.). Then give them these, nicely laid out in playing-card-sized rectangles:

Picture

Hero No. 1
Name:  .......................................................
Description:  .........................................................

Skills:
Constraints: Their four skills must have a mean of 30.
Skills:
1:  ................................................
2:  ................................................
3:  ................................................
4:  ................................................

Hero No. 2

Picture

Name:  .......................................................
Description:  .........................................................
Constraints: Their four skills must have a median of 30.
Skills:
1:  ................................................
2:  ................................................
3:  ................................................
4:  ................................................

Hero No. 3

Picture

Name:  .......................................................
Description:  .........................................................
Constraints: Their four skills must have a mode of 30.
Skills:
1:  ................................................
2:  ................................................
3:  ................................................
4:  ................................................

Hero No. 4

Picture

Name:  .......................................................
Description:  .........................................................
Constraints: Their four skills must have a range of 30.
Skills:
1:  ................................................
2:  ................................................
3:  ................................................
4:  ................................................

Hero No. 5

Picture

Name:  .......................................................
Description:  .........................................................
Constraints: Their four skills must have a median of 25 and a mean of 35.
Skills:
1:  ................................................
2:  ................................................
3:  ................................................
4:  ................................................

Hero No. 6

Picture

Name:  .......................................................
Description:  .........................................................
Constraints: Their four skills must have a median of 35 and a mean of 25.
Skills:
1:  ................................................
2:  ................................................
3:  ................................................
4:  ................................................

Hero No. 7

Picture

Name:  .......................................................
Description:  .........................................................
Constraints: Their first three skills must have a mean of 10, but all four skills must have a mean of 20 and a range of 50.
Skills:
1:  ................................................
2:  ................................................
3:  ................................................
4:  ................................................


You could easily have more or make them easier/harder, but you get the idea. Pupils make their own Top Trumps cards and in the process:

• Get a feel for the differences between the mean, median, mode and range, seeing which ones are most affected by outliers and which average it is best to have higher (in this case).
• Solve more difficult problems involving averages, including finding the numbers when you know the averages, a moving average as data is added, and comparing data using averages.
• Get a reason to experiment and play around with the numbers and not just go for 40, 40, 40, 40 to get a mean of 40.
• Do something creative in class.
• End up with something that they can take home which will remind them of the lesson.

There you have it: Let pupils make the cards, checking their partner's to make sure they don't cheat, cut them out, and play against each other.

But wait, there's more! How about these for great follow-up questions:

• Which cards were the most/least constraining? Why?
• Which could you 'cheat' on? How?
• Next time I'll take those cards out to make it fair. Can you come up with some fairer constraints? Check that the problem you create is possible.
• What about a harder problem? What about an easier one?

As always, suggestions are suggested :)

*UPDATE* Just as I posted this I saw a related post on twitter by @InteractMaths. Would make a great follow-up or plenary to this lesson.

Lesson Sketch: mathematics of a cake sale

Intro:

There is a cake sale coming up and you plan to sell these flapjacks:

How much should you charge for one? Why? Will that make me a profit? How do you know? How much profit would I make? How could I work it out?

Main:

Give pupils the recipe: http://allrecipes.co.uk/recipe/3846/apricot-oat-flapjacks.aspx and either a table of prices or access to a supermarket website, such as http://www.tesco.com/
Pupils work out how much it costs to make a batch of 8 flapjacks and the profit they could make on each one sold.

Extension:

What are the most expensive parts of these flapjacks? Try to replace some things in the recipe with something similar in order to make the total cost lower. How much profit would you make now? How cheap could you make them?
Can you find a different cake recipe that would be cheaper to make?

Further Discussions:

This cake is slightly cheaper to make than the flapjacks:

Should I make these instead of the flapjacks? Why? Why not? What other considerations should I be making apart from price? (Time, effort, equipment needed, etc.)

You may discuss how some people factor time as money using a basic wage.

Lesson Sketch: Relative frequency - A lesson on calculating the risks of sport

Another basic one here. Why do it? Because you can't repeat the 'checking to see if a dice is fair' activity year on year forever. Step one - Ask pupils to put a list of common sports in to the order of how dangerous they think the sports are:

American Football, Football, Gymnastics, Ice Hockey, Swimming, Wrestling, Field Hockey, Skiing, Water Polo, Equestrian, Golf, Skydiving

Step 2 - Show the number of deaths for each sport and proclaim American Football to be the most dangerous!

Sports	Deaths in US colleges in the last 5 years in five years
American Football	55
Football	6
Gymnastics	1
Ice Hockey	1
Swimming	8
Wrestling	3
Field Hockey	0
Skiing	2
Water Polo	1
Equestrian	1
Golf	0

Ask what is wrong with your statement. Why have so many more people died in American Football than any of the others? Get that we also need to know how many people actually took part. Show them the third column of the table along with the formula for relative frequency:

Sports

Deaths in colleges

in five years

Participation in colleges

in five years

American Football

55

2’079’069

Football

6

841’963

Gymnastics

1

57’613

Ice Hockey

1

128’898

Swimming

8

475’953

Wrestling

3

188’272

Field Hockey

0

156’370

Skiing

2

33’974

Water Polo

1

39’968

Equestrian

1

10’489

Golf

0

62’528

Pupils put the sports in order of their danger of death. One thing I like about this is that the data (got from here) is not really adequate for most of them and it gives plenty of room for discussion.


Step 3 - Show pupils this pdf: http://www.uspa.org/Portals/0/MembershipSurveys/memsurvey09.pdf

Ask how we can work out the relative frequency of deaths in skydiving using this data. Pupils will probably go for the 16 deaths over 2.5 million jumps, but for the result to be comparable to the previous task it would make more sense to do the number of deaths over the 32'177 members. This is because all the other sports will be played throughout the year and it is counted as deaths per participant rather than deaths per game. An interesting and fairly advanced conversation to be having with your pupils - YAY!

Lesson Sketch: Using Biometrics to Run like Usain Bolt

http://www.fotopedia.com/items/josegoulao-TbuZScZeYnM

Intro:
Students get into pairs with a stop-watch each. The pairs time each other running 100m, getting split times for every 10m (you will need to have marked the track for this in advance.

Main:
Say we will now try to improve those times by learning from the best: Usain Bolt. Show the pupils graphs of Usain Bolt's 200m World Record run:

 



Students draw a distance/time (and speed/time if pupils have learned about tangents) of their own split times.

1. How is your graph different to Usain's (apart from his being a 200m run)?
2. Using Usain's speed/time graph, describe his actions in a much detail as possible. Give times for each action. Give distances for each action.
3. Why do you think Usain slows down towards the end?
4. Compare Usain's acceleration at the start to yours. What other differences are there between Usain's graphs and yours?

Hopefully from that, students will see that Usain accelerated extremely quickly in the first 10 meters, but continued to accelerate until about the 50 meter mark. From there he tried to maintain his top speed, but slowed down gradually.

(Optional): Show a video of this run. Note techniques: Arms rigid with fast pumping movement, shortish steps, leaning forward at start but upright after a few seconds.

End:
Write down a checklist of what Usain did in the run. Pupils go back to the running track in their pairs and try to do everything on the checklist to improve their previous time.

Loving SMBC's newest comic

http://www.smbc-comics.com/

Lesson Sketch: Nutrition

Act One:

Bring in meals from the school cafeteria.

Which do you think is healthiest/least healthy? Why? What makes a meal healthy? How can we tell?

Act Two:

What information do we need to solve this problem?

http://www.sunshinehealth.net/health-care-forms/food-nutritional-value-search.htm

www.cwt.org.uk/pdfs/eatingwell.pdf

weighing scales

Act three:

Talk to school management discussing their reasons for each meal option and/or

Make posters to help other students make healthy choices 

Follow-up: 

Look at how nutritional needs change for athletes. How could we show this on our posters?

How could we change these meals to make them healthier (may use: http://healthyeatingclub.com/info/articles/nutrients/food-source.htm)?

<Edit>
If you feel the need to convince your pupils about the importance of nutrition, this video does a pretty good job