Math Art – Grade 10 Summative

I wanted to share this task because my students were very proud of their work, and they should be recognized…because they’re awesome.

I want to say first that this was inspired by a post that I saw by Mary Bourassa. So thank you Mary!

The task was to first create a rough sketch of a drawing which they think would contain some parabolas, lines, a circle and a triangle, and then to use mathematical equations in DESMOS to create their good copy.  The rule was that they can only use things that we’ve learned in class, unless they could explain to me the mathematics behind something else they’re using.  I’ll show come cool examples of these situations when I go to the examples.

Once the art was complete, students had to submit two things along with their artwork:

The Equations page:

They had to select 15 equations from their sketch and identify where I can find them.  the 15 equations had to include a variety of:

• Parabolas in Vertex and Factored form
• Lines (horizontal, vertical, and slanted)
• Circles (one was sufficient, and it was allowed to be centred and the origin)

The Calculations page:

Students had to submit some calculations to me:

• Show algebraically the lengths of the sides of one triangle in your artwork
• Show algebraically all of the angles in that triangle
• Determine algebraically the point of intersection of two lines in your artwork
• Determine algebraically the point of intersection of a line and a parabola in your artwork
• Show me how you convert one of your parabola equations to standard form

The Interview:

Finally, students were given the chance to explain their thinking orally.  Individually, they were asked questions like:

• (I’d point to a parabola in their artwork) Tell me an estimate of the equation of this parabola in vertex form, and explain your choices.
• What would the equation look like if I wanted to make this parabola look like (then I’d draw for them a new parabola similar to theirs, but slightly different)
• Describe using transformations the parabola here…(point to a parabola)
• Why did you choose to use the Cos law here (pointing to their calculations)

I’d usually ask two questions about their artwork, and one question about their calculations.

Their summatives were amazing, including all aspects.  This is the actual assignment that we gave them.

Here are some interesting examples:

I love how this student used screenshots to identify where the functions are on the Equations page!

This student went ahead and learned how to create ellipses on her own.  She said that she just assumed that if you multiplied the bracket by a number, it would stretch or compress the circle because that’s what it does to a parabola.  She also felt that it was important to colour in the artwork, so she did it by hand!

This student’s artwork had a title.  “Nonno” – Grandfather in Italian.  When I saw the rough draft, I told her to not be married to the details because she may not be able to fit it all in, but I obviously underestimated her.

A few students exported their image into PAINT and then used the fill function to colour in the final product.  I was amazed at the amount of effort that went into this.

Some were just fun to see…

And others were very interesting artistically.  When I saw the rough copy, I had no idea that the final product would be this awesome…

So thanks to my students for doing these for me, and if you’ve got any ideas or modifications, please let me know!

10 Good Things – A Reflection

So my man Jon Orr tagged me as one of the next people to try this #10goodthings reflection in his post.   Love the idea, for a few reasons.  First, it makes me feel good to say out loud the good things that I’ve done… I think we all need a pat on the back once in a while, and thanks to Mr Orr, I’m about to give myself one.  Also, I love reading others, as they give me ideas…so steal away!

1. I gave myself a new course, MHF, and its going really well.  The students love the open-ended questions and challenging tasks.

2. I invited George Couros to our school, and we started the hashtag #CastleProud which has taken off with staff and students showing their pride in what the school does.

3. I’ve started blogging about my ideas… it really makes me flush out my thinking. In my blog, I’ve organized some 3 act tasks which I think are, and will continue to be hits…Tim Horton’s XXL coffee, and The Ferris Wheel…reimagined.

4. In one day, I met the 4 most influential people in my educational thought: Dan Meyer, Sugata Mitra, Jo Boaler, and Marian Small.

5. I taught an AQ course for Intermediate math…really sparked my love for math pedagogy again…I would rate this as my best experience for the year (professionally).

6. I’ve tried a lot of tech tools…and I’ll keep doing that.  I think my fave for the year is the entire GAFE (Google Apps for Education) platform…so many things that it can do.

7. My students and I have great rapport, still.  They entertain me, and I’m pretty sure I do the same for them.

8. I’ve shared my interest in creating a culture of Growth Mindset with many others around the school and the board…many staff said it was the best PD they’d been to.  I see that as really making a difference.

9.  I’ve remained zen in the face of stress.  Others in my school have asked “Do you guys meditate up the math office all the time?” with respect to how calm the department is, and I like to think that it has something to do with my leadership.

10. I got out to almost every football practice this season! Whoot! 4-3 record in our second year in existence! Go Dragons!

So…go ahead and pat yourself on the back and tell us…what are your 10 Good Things?

Quadratics Summary – Make a Website

So I had my students create a website to summarize their understanding of quadratics.  I was amazed at what they produced.  They used their own choice of platform, but most went with SMORE, Wix, Weebly or Google Sites. This is the rubric that I used:

My favourite part of this process was having students comment on each others work on an LMS.  They gave each other (for the most part) pointed feedback with great ideas on how to fix things.  Then I allowed them to resubmit after taking the suggestions into account.

Here are a few examples, you’ll be able to notice the difference between the students that really took the time to understand and appreciate the visual representations, vs the ones who just learned the rote and regurgitated the information.

Sample 1 – this student kept asking me about decomposition because he wanted a shortcut to trial and error for factoring trinomials… I wouldn’t tell him, so he learned it himself… quite well too

Sample 2 – this website is so well organized and the student did a great job including visuals that are easy to follow.  Actually did a great job teaching everything Quadratic… Take that purplemath.com!

Sample 3 – a simple site, but again went into great detail to explain all of the understanding that she’s gone through.

So that was it… I mean, I can’t say that everyone did amazing work, but I have to say that I was impressed by most of the stuff submitted.  Just amazing what they picked up in class, and also how they went out of their way to fill in some of the gaps that they felt they had.  If you’ve done something similar, please share with me your rubric as I’m not too happy with this one… 🙂

3 Part Lesson – Tim Horton’s XXL Coffee???

I haven’t tried this with my class, but I’m looking forward to it.  If you do try it, please let me know how it went 🙂

Here is a link to a folder which holds all of the images used in this lesson.

And here is a link to the Google Slides presentation with pictures already included .

Act 1 – What questions come to mind?

Show the students this image and ask them what questions they have… I am anticipating answers along the lines of: How much is in each one? Does it go up by the same amount? Is the  XL worth the price compared to the small?

Key talking point – I think its really important to dig into the “Does it go up by the same amount” question if it comes up.  Do you mean height? Does that matter? Do you mean volume? Are these Cylinders? How do we get around the fact that they’re not??? Also, what is measurable…can we measure the volume? What about the radius? What measurements would be more accurate?

By the end of this all, I hope to get to solving the following problem:

“I love my coffee, and an XL just doesn’t cut it by December.  I’d like to have you design a XXL coffee for me.”

Act 2 – The information

Ask the students what information they would like in order to solve the problem.  You can provide as much or as little information as you’d like, and you could have them calculate or estimate the rest.  This is all of the info I collected.  The images are available in the link provided at the start.

Width of the TOP

Width of the BOTTOM

Heights

Prices

Volumes

I think I wouldn’t give the volume measurements.  I’d like them to find the volumes, and maybe I’d release this to check their answers.

Act 3 – So what does this look like?

Here, I’d like to see presentations of each group’s product, with mathematical reasoning as to which one is the best.  As a group, I’d like to come up with the criteria for what the best one will look like, and then we can decide on the winner.

Some of the mathematical discussion I anticipate will be around variables, so for example if two groups look at cost in these ways:

So which one is better? The top graph compares the size to the cost (S=1, M=2 etc), and the bottom one compares the actual volume to cost.  Which one will we use? Do we want to be mathematically fair, or make more money?

So that’s the plan.  Let me know how it works, or what you think of it. Cheers.

Patient Problem Solving – Scaffold Problem Solving, Not Problems!

We want students to do challenging things in mathematics, but we also want them to be successful.  Its a constant battle between scaffolding too much and not scaffolding enough. I’m always worried about not being clear in what I want, and asking students to do very confusing tasks, but I also don’t want to take them through things step by step, because then they’re not thinking about the math themselves.

What does not help?

Telling students what to do.  Giving students exact directions on what to do next.  If we have to do this on a daily basis, then we haven’t taught students how to actually think.  If we have to do this, then we have to scaffold each problem for our students, because each problem is different.  This obviously doesn’t help long term, because eventually we hope that students can do things without us. This involves instructions like:

• create an equation for this sentence
• measure the slope of the line

What does help?

Scaffolding thinking.  Using things like the Mathematical Processes to help our students think about big picture mathematics.  Having students do things like reflect on the reasonableness of their answer is useful in every problem.  Having students understand that we can represent mathematical models in various ways also helps in a variety of problems. Another way to use scaffolding is to scaffold in time during the lesson to have students discuss their own next steps.  Perhaps its a few minutes to talk about what should be done next.  Perhaps its some time soft students to look up vocabulary words that will help them solve a problem.  Here is a great summary of a few of these strategies.

Patient Problem Solving

In the end, its simple.  Help students think so that they can solve their own problems.  Don’t help them solve specific problems.  Long term, that’s not helping. If we scaffold each step, they will never be patient enough to think through something on their own.  On the other hand, if we scaffold thinking strategies, they will take the time to reason through their thoughts and work with others to make sense of the mathematics that we’re trying to teach.

The Ferris Wheel Problem – What info do you need vs. What info can you get???

I wanted to get my students to model periodic behaviour using Trigonometry.  This lesson did that, but it got us to talk about something much more mathematically important: What information is needed, but also what information is attainable?  For example, my students wanted to google the “Central Axis of the Ferris wheel at Chicago Navy Pier”.  This information is necessary to solve our problem, but unattainable in these words… I included a video with this discussion, and had some really great conversations with students about what information we can actually expect to find out there, and what math we can use in order to manipulate the information we find, into what we need.

Here are the slides I used.

Act 1: I started with just this clip of the Navy Pier Ferris Wheel from Chicago and asked the students what questions they had:

Not the most effective Act 1 I’ve ever had, but it got them to think about things like:

• How long are you on the ride?
• How tall is it?
• How high off the ground do you get on?

But the question that got my attention, and that drove the rest of the class was: How far can you see from up there? So we made up the question “You can see US Cellular Field from a height of 100m.  How long do you have to take pictures of it?”

Act 2: We brainstormed for a long time the information needed.  I’ve included a video of the discussion here, but I think the coolest thing that came out of it is that we learned to translate math language into Lehman’s terms.  For example, my students wanted the “central axis of the Ferris Wheel” and I asked them if we could google it.  They all agreed with a “no” and told me I was ridiculous.  I agreed, so we tried to find what it was that we could google about the wheel.

Also, towards the end, a student asked if we could google the answer to “You can see US Cellular Field from a height of 100m.  How long do you have to take pictures of it?”, and I really appreciated that comment. Lots of good discussion came from it.

I finally gave them very little information that I could scrounge up, and it was this little slide:

Students moaned and groaned a bit, because in the video you may have noticed that they wanted things like: Radius, diameter, min heights, vertical shift etc… and none of that is given… they had to use math to interpret all of this information to make it usable.

Act 3: Use your model, and interpret whether it makes sense or not.

So that was it from me, the rest was on them.  They actually came up with very innovative ways to determine the second time when you’d reach a height of 100ft.  Here’s some of the work they produced:

I actually tried to interrupt them a few times…but to no avail…they wanted to figure it out themselves.  They make me want to keep teaching…

Let Students Impress You

I just had a conversation with a student in my class that made me really happy.  I got a student to believe that her figuring out information, made her a lot happier than me giving it to her.  I got her to impress herself, and to believe that she can impress me.

Completing the square is part of our curriculum, and we teach it as just a way to change standard form to vertex form when it comes to a quadratic relationship.  So after doing a lot of fun tasks around modelling situations that end up being quadratic, we have to … complete the square.

Bottom line is that I could easily tell my students that in order to complete the square in $x^2+bx+c$, you take half of the “b” value and square it, but what’s the point???

My students have worked with algebra tiles before, so I just ask them to take $x^2+4x$ and arrange it as close as possible to a square. So for a while, their desk just looks like…

After a bit of thought and some talking and arguing…it eventually turns into…

And then I just ask them…”what’s missing to make it a square?” and they’ll very quickly show me this:

The Side Conversation

So while this is going on, one of my students who, before this semester, had a tough time with math class, starts a conversation with me:

Student: (Quietly) Sir…is it four?

Me: What do you think?

Student: (Louder) I think it has to be four…

Me: Could it be anything else?

Student: (Confident) Nope! It has to be four.

Me: Ok then…

Student: (With a big grin) I’m so smart!

The Cynicism

So into the second question I asked them, another student at her group says…”I’m pretty sure that its just “b” divided by two squared”, and the girl asks “Sir…why wouldn’t you just tell us that???”

The Teachable Moment

Me: How did you feel when you figured out that the answer is 4?

Student: Awesome!

Me: So what kind of person would I be if I took that away from you?

Student: Awwww… Thank you sir!

The Bottom Line

We have many answers… answers that can impress students… make them say ooooohhh and aaaaaahhhh, but why??? Why take those moments away from them? If we give them the chance, they can make their own moments, impress themselves, and most definitely impress us.

Cheers.