Building understanding of the equals sign

I recently read a paper called the Importance of Equal Sign Understanding in Middle Grades, where it is argued that most middle-schoolers view the equals sign as a kind of signifier for "That's where the answer goes" (3+4=?), rather than understanding the deeper, relational meaning (3+?=3+29+10 or y=2x+1), and that this significantly hinders their understanding of simple algebra problems.


I still see this misunderstanding whilst currently teaching college-age students. This is worrying.

Ways that understanding can be taught:

• Dave Orritt has created a lesson with resources looking at questions of the form 3 + __ = 2 + 9, etc. of varying difficulty.     I would have liked a little progression such as:
    5 + 21 = 13 + __
    5 + 21 + 154 = 13 + __ + 154
  
  
• Nrich has a nice problem that involves changing equivalences, so that even before you get on to generalising, you can look at how the equals sign still applies as you move around the grid.
  
  
• Here are two (first and second) similar web-apps dealing with balancing equations. They are made to be used to help learn solving equations, but you could also just let students try things out to see what happens to the equation.
  
  
  Explore linear graphs starting with finding values of x and y that make the equation x + y = 5 (or any other) true, then plotting them all as points on a graph at the front of the class. (What happens to x as y increases? etc)
  
  
• Building up equations. Can't remember where I found this, but its the best version of its like that I've seen:
  
  
  
  Which could then be followed up with students building their own similar webs.
  
  
  
• Practicing simple, worded and otherwise, formulae; including substitution of different variables and rearranging. You could also ask if different forms (e.g. F=ma, F/a=m, Fm=a) are always, sometimes or never the same and to justify their answers. 


• Equation chains - e.g. 3 + 2 = 9/? + 2 = (2+?)/3 + 2 = 25/?


• Hands-on equations or something similar

 

How would you do it?

These are some initial suggestions but I'm wondering if anyone has any other ideas.

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.

WCYDWT: Bouncing Ball

Editted footage from: http://www.youtube.com/watch?v=sKJegbjS4N8
Would like to film it myself with better quality (and a measuring stick) in the future.

Lesson Sketch: Archaeological Classification

I'm currently in the process of making a series of lessons based on archaeology. Its been really hard going and I'm only marginally happy with the results. I'd really love some feedback about a lesson, which I'll post below. How can it be improved? How can I get across the lesson activities more concisely? Is there any way I could make the lesson activities more intuitive to students?

Lesson Plan – Classification

Background information: The question of classification of finds is a very important one in the study of Archaeology. Pottery, for example, can be classified in terms of type of vessel, date of manufacture, place of manufacture, and style. For these classifications to be meaningful for analysis they must be reliable and consistent, so that different Archaeologists would classify objects in the same way. One of the simpler ways to do this is to give inequalities in terms of measurable properties of the pieces (e.g. Diameter of base).
Objectives: Pupils define inequalities that distinguish between the different types of pottery. Pupils use these inequalities to see what type of pottery an artefact is.
Duration: 1 hour		Prior Knowledge: Cartesian co-ordinates, equation of a straight line
National Curriculum Links: 3.1e Linear, quadratic and other expressions and equations: They solve inequalities in two variables 3.1e Linear, quadratic and other expressions and equations: They use algebraic and graphical methods to solve simultaneous linear equations in two variables.						Level: 8 7
Keywords: Classification Pottery Inequality Consistency			Resources: Classification – Resources		Equipment needed: ·         Projector or interactive whiteboard ·         Rulers
Times: 20 minutes	Introduction: Give out copies of the examples of different ancient pottery (resource no. 1) Pupils work in pairs assigning one of the four types: Plate, Bowl, Pot, Urn to each piece of pottery. Display the examples of different ancient pottery on the projector. Go through the pieces, recording and discussing any disagreements. Ask the students to imagine that they are archaeologists working on different sites and we would like to compare the finds on each site. Ø  What problems would arise from disagreements about the different sorts of pottery? How could we solve them? What properties does a plate have?
25-30 minutes	Main Activity: Explain that if we can describe a type of pottery by things we can measure about it, like height, then we could consistently classify objects. Ø  What properties could we measure with this pottery? Display the heights of known pottery types (resource no. 2) and the rim diameter of known pottery types (resource no. 3) Ø  Can we say anything in general about the heights/rim diameters of plates/urns? Why is this not enough to define what makes a plate? Display the graph of heights vs. rim diameter (resource no. 4). Explain that we are now looking at both properties at once. Ø  Is it now possible to draw a line to separate any of the pottery types from the others? Invite a pupil to draw this line on the board. Go through the steps needed to draw this as an inequality. Show how to shade in the unwanted side. Ø  What are the properties of the objects in the wanted region? Is this reasonable for a plate? Hand out the graphs of heights vs. base diameter and base diameter vs. rim diameter (resources 5 and 6). Explain that we can use these graphs to separate pots and urns from the others, but we may need more than one inequality. Pupils find these inequalities. Ø  What are the properties of the objects in the wanted region? Is this reasonable for a pot urn? How can we tell if something is a bowl?
10-15 minutes	Plenary: Go through the pottery finds slides (resource no. 7). First show the estimated dimensions of the artefact. Show the artefact once pupils have decided what type of pottery it belongs in. Ø  What type of pottery is this? How did you work it out?
Differentiation, extension material and further reading: Ø  How would you change the inequality to describe all the pottery that is not a plate? What happens if an artefact lands exactly on the line? Ø  How can we deal with disagreements about which line to draw? Does a ‘best line’ exist? Could we define rules to draw this ‘best line’? http://antiquity.ac.uk/projgall/rosania/index.html Shows how a similar technique was used by looking at the relative amounts of certain rare elements to locate the origin of obsidian artefacts.
Links to other subjects: Plant classification				Other Comments:

Graphing Stories: Gran Turismo

I may have missed the deadline for Dan Meyer's graphing stories extravaganza, but an idea struck me last week for some speed/time graphs.

What I like about this:

• It is something that people have lots of knowledge about, even if they are not particularly into racing games, that they can bring to the discussion on these problems.
• The timer and speedometer are already there so you can verify your answers easily.
• On the video for Gran Turismo 4 it shows the track. I would pause the video at the start and show what part of the track is coming up and invite guesses to the shape of the graph before even showing the video once.
• It allows the decent extensions of acceleration/time graphs and distance/time graphs (although units for that would get a little difficult, getting the general shape would be fairly straightforward)
• There are at least 30 of these tests all filmed on youtube.

Possible shortcomings that come to mind:

• The biggest problem I have with these videos is that the counters and map aren't shown at zero seconds. Didn't the Gran Turismo programmers know that I'd be using their game to teach maths?!
• The videos don't really force us to use our intuition and some pupils may just end up frantically recording the speeds without much thought. (This is negated if you get a guess before even showing the video)

Here are the graphs I get for the video above:

Lesson Sketch: Solving problems graphically

Here is a pic I recently used in class, which was taken from an idea by Dan Meyer:

The question of course being "Have I scored?"
I had four different pictures. We went through one together on the board. We tested our calculations by watching the full video. Each pupil then tackled a random choice of the others. Difficulties as always arose trying to be anywhere near accurate on the interactive whiteboard, but aside from that the pupils seemed very engaged and eager to contribute in the lesson. Admittedly that isn't too difficult with the pupils in this school, but what I really enjoyed was the way some of them seemed so shocked, firstly that an equation could actually be used for something, and secondly that it worked.
One touch I like about the image is that you don't need any extra props to establish scale.

Notes for improvement:

• I think I rushed through the early example, afraid that they wouldn't get it or would lose interest without my support. I needn't have worried as the pupils rose to the challenge very quickly and a little more patience on my part would have encouraged pupils to test ideas out with each other and come to an answer independently of me.
•  On the pictures the pupils worked on themselves, most of the kids were about 5 mins from having their answer when the bell rang. I panicked and showed them the videos revealing the answers anyway. Doing it again I would probably allow two lessons for the topic and would just take my time. I guess I'm still getting used to these 35 min lessons.