Transforming functions with Geogebra *UPDATE

Here is a quick transformations of functions app I made with Geogebra for my C3 class. The aim is to use transformations of standard functions (including absolute value) to find the function shown. It's too big to fit properly here, so follow this link to it.

*UPDATED

Also, transformation starter activity:

 

and answers:

#MakeoverMonday - Modelling cost per minute

 This is my attempt at this week's #MakeoverMonday. This is the original problem:

I would love to know what you think, so give it a read and let me know. Ok, lets get started!

Introduction

Tell a story (optional): "This is me travelling in South America.

I'm on a really tight budget (hence hitching a ride on top of a truck!).

Here are some pictures I'd love to post on facebook from my trek to Machu Picchu:

So I go to the nearest town and see these two internet cafés

Pose the problem: Which should I choose?

What factors might be important to consider (Cost, internet speed, how nice they are inside, etc.)? How could I find the information needed to make a smart decision?

At this point give them the image from the original problem, "This was all the information I could find:"

Subtext: the story introduction is something I've put in because one of my goals this year has been lesson personalisation. Its something that has paid off great dividends in terms of my relationship with many of the pupils.

Also, storytelling is supposedly hard-wired in to all humans, which means that the problem is framed in a relatable and memorable manor, even if the pupils are not capable of relating to the algebra on the horizon. Its optional because it may not suit your style or there may be classes who may just get confused/distracted with the extra detail.

The problem is posed simply and honestly. There are no 'write an equation' or other scary sentences, and it feels like this is information that you would readily be able to get from each internet café.

By the way: to make the problem more applicable to internet cafés I would change the <per hour> in to <per 10 minute slot>.

Building in the Maths

Specific: Some bright soul may say that it depends how long you're going to be on the internet for. Cement this idea by asking:

• Which is cheaper if I'm online for 1 hour? 2 hours? 3 hours? 4 hours?
• Will that café always be cheaper?
• Try to find an amount of time that would make 'internet action' cheaper.

General: Scaffold the writing of equations (if needed):

• How are you working out the cost each time (for each café)?
• Could you write what you're doing in words?
• What about in symbols?

Representing on a graph:

• Could you plot these equations as lines?
• What does the point where they meet represent?

Subtext: In the original problem, part d is a significant step down the ladder of abstraction and is much easier to solve than a-c. I've switched it around, going from specific cases to the general equations.

I've added my way of scaffolding writing equations (where you write down what calculation you are doing in words first). I've found it really helps pupils to connect algebra to what it actually means.

You could also have a 'take a guess when they will cost the same' section before moving between the specific and general case. This may improve motivation for the general case, but in this case pupils may feel its easier to use trial and improvement (thus demotivating the need for the general case).

Follow-up and Extension

Offer similar problems: e.g. Dueling Discounts or Stacking cups or just written questions.

Extend the problem: Put the question backwards by giving the costs for various amounts of time online and get the pupils to work out the price-scheme. You can also extend this by having a messier version of the same problem.

Subtext: You have to be careful here that the similar questions are not so similar that they become mindless. I like the duelling discounts and the toaster regression for this as it is not immediately obvious that the same technique would be applicable here. One of the 'big ideas' I would like my pupils to learn about maths is the portability of problem solving techniques in to different types of problem.

I used the classic extension method of reversing the question here. Offering a problem with messy numbers or without perfect correlation is another portable extension tool I often use.

Interactive: Live Distance-Time Graph Creator

For a recent gifted and talented session I ran, I created This Excel File, which creates a distance time graph live. I gave pupils a pre-made distance time graph and pupils had to recreate it by walking across the room. I used it last week and it worked really great; pupils loved getting up and trying to time their movement. I'm sure it could be used in plenty of other ways too.

 

Setting up the room

• Make a clear path along the length of the room.
• Lay strips of tape in parallel lines spaced one meter apart across the length of the path.

Using the file

Open the file and click 'Enable Content' at the top.

Sheet1 - Creating the distance time graph

• Press reset to delete old values and set the view.
• The 'S' Column contains the timing of the 'Original' line. Change these for a different target graph. (Edit: Times have to be written in this format to work - hh:mm:ss)
• The 'T' Column contains the distance values that are shared between the 'Original' graph and the pupil graph. Change these if your room is a different size.
• Press prime to activate the distance time graph creator.
• Once primed, press enter (on the numeric keypad) every time the student steps across each line of tape, and a new point on the graph will created. (Note: first point created is at (0,0) and also starts the timer)

Sheet 2 - Working out speed
 

• 'View' sets the view
• Each button below this creates the triangles needed to calculate average speeds, but currently must be clicked in order (from top to bottom) for them to work. I will fix this eventually; at the moment the file is limited to my one use session, but I'd love to expand this to be a more multi-purpose graphing tool.

Resource Shout-Out: Modelling Epidemics and an Investigation that Leads to a Proof of Pythagoras'

For the start of this term I have been asked to make a couple investigation lessons for each year group. Nothing too difficult, but just something that pupils could get on with after being talked at about syllabus, expectations, homework policy, etc.

Here are a couple of GREAT activities that I just adapted slightly for our classes:

• Investigating Epidemics from nrich.maths.org I just removed the 'counter plague' to spend more time letting groups try to improve the model and try it out on the class.

• Sloping Squares by noycefdn.org (pdf file)I really like how you end this investigation with a² + b², on the cusp of proving Pythagoras' Theorem. There seemed to be too big a leap for me between question one and the algebra in question two, so I added a middle question: 

  "What sizes of square is it possible to make (below 25)?"

When you look at the possible areas, you can see how they can all be made by adding two squares together and hopefully this will help bridge that leap in to the general case.

  Possible areas: 2, 5, 8, 10, 13, 17, 18, 20, 25

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.