Pumphrey's Math

Patterns, everywhere


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Visual Patterns and Coding – Part 1 – Linear Relationships

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I have been running a ‘visual pattern’ every week with my 6th grade (pre-Algebra) classes. You can read more about this here.

To bridge the gap between pattern and function and following an online course I took with Rice University, I have started to introduce some basic coding. Python in particular. Even after one lesson of using coding and graphing, I have been able to have rich conversations about the differences between functions, input/outputs, the shape of a graph and the y-intercept. Here is the process I have taken them through:

Part 1: Have the students run through a basic (linear) visual pattern (from visualpatterns.org) using this sheet and reviewing using this slide:

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The nth term for this pattern is 4n+1.

Part 2: I now challenge them that we can create a calculator for this pattern using the Python coding language. I use the free python interface CodeSkulptor (from Rice University) to do this. I take them through step-by-step with some great conversations about functions and inputs/outputs.

The nice thing about CodeSkulptor is that when you hit the save button, it creates a brand new URL meaning that each student will have their own URL to post and share.

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They can then change the input and see clearly what happens to the output.

(Note: Lines with # are ignored by the interface)

Part 3: They then go to the Desmos Online Graphing Calculator and input the function y = 4x+1 to confirm or deny their prediction for the graph shape, from the start of the exercise. This is a great opportunity to talk about ‘step zero’ (as well as step -10 etc.) and why they graph is the shape that it is. I feel it is also important to stress the difference between 4n+1 as an nth term and y=4x+1 (which includes everything in between).

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Their homework is simply to follow the steps with a different (linear) visual pattern and to share their CodeSkulptor URL’s and Desmos screenshots on the class’ wiki page.

For student assistance I created this video:

Where Next?

There are two main places that I would like to take this:

  • Exponential functions
  • Inverse functions

I’m really excited about where this journey will take us. My hope is, that as these students start Algebra proper, next year, they will have a strong sense of functions graphs and their connections with patterns and geometry. Here goes…..

Have you done anything similar? I would love to hear your ideas/thoughts in the comments section, below.

 

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Ten Webites I Check Before Starting Every Unit

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There are many excellent resource sites out there with inspirational teachers making great resources. However, when it comes down to it, there are only a handful of websites I check on a very regular basis. To make it to this list, I am generally looking for tasks that:

  • Are ‘low floor/high ceiling‘ giving access and challenging everyone at their own level
  • Have multiple approaches, giving students room for creativity
  • Are challenging and hence interesting
  • Will give me permission to teach skills
  • Make skill practice, more interesting
  • Have a story for students to follow

There are many activities out there that fit these descriptions, but there is only so much time in the week. During breaks I have more time to explore but these are the sites that I can check quickly with high yield for your classroom (in no particular order):

1. Mathematics Assessment Project – A website produced from a partnership with the Shell Centre (University of Nottingham, UK) and the University of California at Berkley. Here you will find tasks that will help the common core standards come alive in your classroom. Full of rich activities that encourage discussion and investigation.

2. Standards Unit – This has been around for a while but is still one of the first publications I look at and is close to my picture of what mathematics education should like. A rich, dense set of activities that will give you great ways to work on various skills and topics.

3. YouCubed – A relatively new site that has come out of Prof. Jo Boaler’s (Stanford) efforts to encourage the ‘growth mindset’ in the classroom. I am interested in anything that comes from the idea that anyone can be good at math.

4. Open Middle – I came across this site relatively recently and am sad that I didn’t find out about it sooner. it’s tag-line is: ‘Challenging Math Problems Worth Solving’ and tips the proverbial hat to the school of thought that says that you don’t need to have tenuous links to real world problems in order to get buy-in from students. It appeals to the problem solver in all of us.

5. Emergent Math – With its routes in Problem Based Learning (PBL), emergent math is useful not only for those who want to tear up the textbook and start again with an integrated curriculum, but also for people who just want great projects that they can slot into their established curriculum. Lots of links to sites not mentioned here so worth a look.

6. Mr. Barton Maths – Lots of great resources and activities that will make your classroom a more interesting place. Enough said.

7. Tarsia – Sometimes students just need to practice. There I just said it. But using Tarsia you can avoid ‘death by worksheet’ and get students to practice without really realizing it. It also great for discussion and you can tell very quickly if students have answered everything correctly or not. For more, see Why I Love Tarsia.

8. NRich – Great site for ‘low floor/high ceiling’ problems that will challenge anyone in your classroom. Students may like to explore this outside the classroom, too. Sortable by topic and I believe they are coming out with a common core curriculum map, quite soon.

9. Dan Meyer 3 Acts Spreadsheet – Act 1 –  You show a video or picture prompting discussion, prediction and estimation as well as the all important step of coming up with the variables that are to be investigated. Act 2 – Students get the information they need to solve the problem. Act 3 – Once students have solved the problem in various ways and presented you show them the solution. There is a lot more to it than this and to pull these lessons off well is a true art form. But the only way to get better at these is to try them. So try them! More info here.

10. Mathalicious – Math lessons based on the real world problems. $185 for 12 months subscription (I think currently they are also doing pay-what you can) but well worth it. Excellent for creating the need for the math you teach.

It takes me around 90 minutes to trawl through these websites at the beginning of a unit but is well worth it when it comes to lesson planning and I know half of the activities I am going to do already. As with any of these activities they will need to be (and should be) adapted for your classes and situation but they provide an excellent starting point from which to plan.

What websites are a must-check when you are planning a unit? Leave a comment, below.


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Walk the Line – Adding and Subtracting with Negative Numbers

I wanted an activity that would give students a deep sense of adding and subtraction involving negative numbers. Both. At the same time. I figured the best way would be for them to actually walk through what happens to a number when it goes through these operations; so began ‘walk the line’.

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The idea is simple, have students walk through various sums increasing from adding two positive numbers, going through to subtracting two negative numbers, reviewing each time.

They will first need to stand in a line, then using sidewalk chalk, draw a zero on the floor and draw out a number line to the right and left of where they are in the positive and negative direction.

Here is the general dialogue that happens:

  • I am going to call out a sum and I would like to see how quickly you can get to the result. You start with the first number I call out and are permitted to move before I have finished calling out the sum.
  • 3 + 2 (call out slowly to give students time to move)
    • Review Questions
      • Now, what did you do when I first said 3? – Run to that number
      • What did you do when I said “add”? – Got ready to run away from zero
      • What did you do when I said 2? – Ran two spaces away from zero
  • Next:  5 – 8
      • Review Questions
        • Now, what did you do when I first said 5? – Ran to 5
        • What did you do when I said “subtract”? – Turned to face the zero (Why? etc)
        • What did you do when I said 8? – Ran 8 spaces and ended up at -3
  • What about  5 – ( -8)    (call out slowly)
    • Review: What was different when I said subtract -8 instead of 8? – Had to go the other way (Why? etc.)
  • At this point I bring student thinking together and explicitly clarify the rules:
    • Whatever number I say first is where you start
    • If I then say “add” you face this way (pointing in the positive direction) and if I say “subtract” you face the other way (pointing in the negative direction)
    • If my second number is positive you walk forward by that amount, if my second number is negative you walk backward by that amount (for example: 4 would be four steps forward, -4 would be 4 steps backward)
    • Let’s try this out
  • 3 – 8 (call out slowly to give students time to think and move)
  • -4 + 10
  • 2 + (-5)
  • -9 + 12
  • 2 – (-5)
  • -4 + (-7)
  • -6 – (-10)
  • etc
  • Then return to classroom and do the same sort of thing but students write (just) their answers on their mini whiteboards. We review each time going through the 3 step process.

This activity really worked and students were doing this all in their head by the end of 50 minutes which is what I was aiming for. There was no separation of addition and subtraction or positive and negative numbers. They were just different points on the number line and different ways to move.

How do you introduce adding/subtracting negative numbers?


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Gapminder is Awesome

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I wanted something that would open up the world of scatterplots to my statistics students; something where they could really get a sense of correlation and causality. I decided to do a project based around the fantastic GapMinder World and it payed off.

First I showed this video of the master Hans Rosling at work with the graphs his foundation came up with.

I then gave these instructions to my students:

You (and max one other person) are to prepare a 3 minute presentation on a GapMinder graph of your choice.

Instructions:

  • On a computer go to the gapminder website by clicking here
  • Play around with the explanatory and response variables until you find two that you think show some sort of relationship
    • If you are struggling to find variables with a link, click ‘Open Graph Menu’ and play around with graphs that have already been created.
  • Your presentation must include answers to the following questions
    • What are your explanatory and response variables?
    • What is the link between variables at the start (before you click play)?
    • What do you notice happens over time?
    • Are all the countries close together or more spread out? What does this mean?
    • Are there changes to any particular country that are of interest to you?
    • What if you isolate by continent? Are there any changes that are of interest to you?
    • Is there anything else that stands out with your graph?
    • Are there any outliers to the trend?
    • Does this graph bring up any other questions that you would want to investigate further? What information would you need to answer these questions? Is this information available?

Your presentation must include the time series animation (when you press play) as well as PowerPoint slides using screen shots of points of interest.

You will be graded on:

  • Content (out of 6)
  • Presentation (out of 4)
What went well
  • This was a great way to get across a sense of scatter graphs and will be awesome to segway into taking about correlation and causality.
  • This was enjoyed by the students and really got them thinking about statistics and global affairs
  • It was good to give specific questions for the students to answer. In my experience just saying ‘present for 3 minutes on a graph of your choice does not give great results’
Even Better If

Next time I do this, I think it would be good to model what an excellent presentation looks like. I missed a good opportunity to teach this skill.


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Desmos is Awesome

I love Desmos. I love how user-friendly and clean the whole thing is. I want my students to love it too, so I used the first lesson back after midterms to let them play.

The Aim

For my students to get a feel for the shape of various functions and relations through using Desmos to create a piece of art. (CC Standard F.BF.3)

The activity

  • Students take a look at http://www.desmos.com/art to see what is possible just by typing in equations (Great hook)
  • I explain to students that they have this lesson and a homework to come up with a piece of art of their own using desmos.com/calculator.
  • I have a sheet ready with some example functions (linear, quadratic, circular relations, radical and rational) for them to use if they are struggling. I also introduce the idea of sliders for them to use.
  • They have around 50 minutes plus a homework to come up with a piece of art of their own.

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Some of the resultant artwork

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What went well

  • Any lesson where students are crying out for the Math is a good thing. It was amazing to be asked how to draw a smiley face using a parabola and domain and range and how to draw circles and ellipses. I had one student ask how to do a ‘diagonal porabola!’ I had to look that one up.
  • Students were constantly engaged. Sometimes frustrations got the better of them and they needed some encouragement to keep going but generally, the lesson went really fast.
  • It was great for all abilities. Students that normally struggle got the chance to play around with linear and quadratic functions, helping them to understand what changing the numbers did to the graph.

Even better if

  • This was too early in the year to do this lesson. I would like to do this next time at the end of the year when students had more functions and tools at their finger tips. I did like how it cemented the need for domain and range, though.
  • This lesson is leading into our quadratics unit. Going back I would have really liked to focus far more on parabolas so that our next lesson on Vertex Form would make sense from the start. I still think it will help, I just think I may have missed an opportunity to go deep rather than broad.
  • This lesson relies heavily on technology. Being a Bring-Your-Own-Device school, some students had tablets that were very tricky to use (this also happened with Khan Academy). I may have to rethink how I do this and use it possibly for homework.

Student Reaction

Here are results from a mini survey I did at the end of each class

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Students also said:

This activity allowed me to visualize what adding variables does to the shape of an equation.

I loved the creativity involved with it, but also the brain work involved when trying to make different shapes and move them around.

Technical difficulties were frustrating, but I realize this is something that is hard to fix.


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How to fake a Jackson Pollock

Convergence (1952) by Jackson Pollock (from about.com)

It’s the end of summer school session 1 and I wanted to give my students a taste of the beauty and power of fractal geometry. Rather than just play a TED talk (although this helped with the introduction) I wanted them to live through the process.

I know that fractal geometry has been used to spot fake Jackson Pollock paintings and thought it would be fun to look at the method behind this and to see if my students could use what they have learned to try and create their own ‘Jackson Pollock’. Here is a summery of what we did, spread out over a 2 day period (in between normal curriculum activities).

  1. For a link from Geometry, semester 1 work, we used this nrich task to start them questioning what a fractal was.
  2. We had a discussion on where you might see such shapes in the real world, e.g. lungs, coastlines, cauliflower etc.
  3. We watched this TED talk from Mandelbrot himself (recommend using subtitles with this, he can be hard to understand)
  4. Before getting crazy with the paint, we watched this video on how Pollock painted his work and how people have tried to replicate the motion
  5. Time to head out to the front lawn of the school where students were greeted by a large canvas and a lot of paint where they were tasked to try and create their own Jackson Pollock, but not before reading the following quote from Nature Notes to help them think through the process, rather than just splash a load of paint around.

‘There were two reasons to suspect that Pollock’s paintings might obey fractal geometry. Moving around a large canvas laid on the ground, the artist let paint fly from all angles, using his whole body. Human motion is known to display fractal properties when people restore their balance, says Taylor, and films of Pollock seem to show him painting in a state of ‘controlled off-balance’. Second, the dripping and pouring itself could be a chaotic process. 

“Pollock was in control,” says Taylor. The large-scale fractals are a fingerprint of the artist’s body motion, he notes. “But the small-scale fractals are also to do with his choices — his height over the canvas, the fluidity of his paint, angle and force behind the trajectory, and so on.”’

The real magic was when we took a photo of the canvas and zoom in several times to see if we have replicated the infinite properties that Pollock created so well. Here is a video of the class starting the painting (and getting pretty covered in paint in the process).

Here is the end result and the zoomed-in versions.

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I appreciate we’re not going to win any awards with this art but from a geometry point of view, I was really happy with the results and so were the students. The zoomed in photos do not look all that dissimilar to the painting as a whole.

I will be, of course, asking the school to display this somewhere. There is part of me that hopes they decline so that I can take it home with me. Apart from anything else, it was great fun!

A huge shout out to local artist, Quincy Owens, who helped me to set this up and provided all the materials we needed.

Have you done something similar? Any suggestions on how I can improve on this for the future?


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Escape the Worksheet: Sidewalk Chalk

Mathematics students need to practice mathematical skills. That much is certain. What I am trying to do in my classroom is to get away from the here-are-20-questions-go activity that puts shivers down the spine of many students. So how do I still help them to practice but in a less monotonous way?

A simple idea, now that the weather has improved, is for the students to do the same problems but using sidewalk chalk to decorate the school’s pathways with beautiful math. There is something in this that appeals to the 3-year-old in all of us.

The lesson objective was clear: To prepare for the quiz on 1) Solving triangles and 2) Using the unit circle to find trig ratios. We wandered outside and went for it.

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How it went:

  • Most students enjoyed the lesson with a handful opting to carry on with a paper worksheet. At the end, students fed back that they enjoyed working outside.
  • I loved the instant assessment. It took very little time to see what was going on in the students minds. It also kept the accountability high. It is difficult to a student to fake writing their work on a sidewalk and check out.
  • It would have been useful to have a little more structure to the activity than just going through the worksheet but on the sidewalk. Perhaps assigning roles, for example, scribe, coordinator, calculator.
  • The sun was hot meaning that students became lethargic towards the end of the lesson. Choose a shaded spot if possible.