3D Perspective Drawing

Today’s lesson with Year 7 looked at front, side and top perspective drawings of 3D shapes. The moment of inspiration hit me when I woke up this morning (which is an improvement over the more common 10 minutes before the lesson starts), but I decided I wanted students to create their own perspective drawings using their imaginations, then turn them into 3D themselves.

This is a lesson in three parts:

Task 1: Warm-up practice

I expected that my students had seen drawings like this before, but I was unsure how confident they would be. So, I gave each pair of students an arrangement of blocks to draw from the three perspectives.

I had one colleague comment in the morning that it was nice to see me playing with my toy blocks…

Happily, they found this pretty straight forward. Some students finished quickly, so I had them swap their block with another pair so they could draw another object.

Task 2: Drawing perspectives

I asked students to imagine their own 3D object made from cubes, then draw front, side and top perspectives of it. To give them an idea of the type of object they could create, I showed them this image:

That is a green car I created very quickly this morning before school. Not, as some students claimed, two trees next to each other…

It was interesting to see the way different students approached this. Some immediately had a creative idea, and were very carefully plotting out squares on their grid paper. Others got more excited about drawing the picture they wanted, but didn’t worry so much about making sure their shapes stayed within the grid pattern. And still others just wanted to finish the task and thought a square would be good enough.

Task 3: Creating 3D shapes

This morning I looked for a website that I students could use to create their shapes easily. After trying a few sites, I found Voxel Builder.

My idea was that I wouldn’t have to tell students if their perspective drawings were correct. They should be able to move their own 3D model to the different perspectives and see if they matched what they had drawn. If they didn’t, then something needs to be fixed with either the drawings or the model, or both.

I haven’t had much chance to explore Voxel Builder, but it seems to have some pretty neat features, including exporting to 3D printers, printing 2D templates for 3D paper models, and animation.

I had my kids export images of their creations. Here are some of them:

 

Rotations around a point

In introducing rotations to my Year 7 class, I had them create a… thing. “Foldable” isn’t really the word I’m looking for here. I think it’s better described as a “spinable”.

Anyway, it looks like this…

…and this.

The main idea is that students aren’t just told what a rotation is, and they aren’t just shown, but they actually create the rotation themselves.

To do this, each student will need:

  • An A5-ish sheet of paper (or half a US letter sheet will do).
  • An piece of tracing paper half the size.
  • One of these pin things. I always called them “split pins” growing up, but I think they’re actually called paper fasteners.

Get students to fold their paper in half, and draw any picture they like (school appropriate, of course) in one half. I only gave them 30 seconds to draw a picture, because I didn’t want them spending the whole lesson on it, and I think a simpler picture is better than this. Of course, many of them took longer than that just to find something to draw with…

Next, unfold the paper and cover the picture with the tracing paper. Attach the two sheets together with the pin. I emphasised the point to students that they could put the pin wherever they wanted, not just the center. I wanted there to be a variety of pin location, so we could see how that affected the image that was produced.

Trace over the picture. I found pencil, rather than pen or marker, works best for this. The colouring is optional.

Finally, glue the back of the paper and fold it in half again. This just stops there being a possibly rogue pin sticking out the back of the sheet.

And we’re done! I also had students write the words “Rotation around a point” on theirs and glue them into their workbooks.

What I like about this is that it emphasises what it means to rotate around an origin. When students were working on questions from their textbooks, it made explaining the origin a whole lot easier: that point doesn’t move, because that is where the pin is. Everything else moves around that.

I also like the way that each student was able to put their own “spin” (get it? Sorry…) on this task. Students could see a whole lot of examples by seeing what other kids did.

Rotations are an interesting topic to try and teach. One battle we have as teachers is finding ways to explain concepts that seem obvious in our own heads, even though they’re not. It turns out that adult-with-mathematics-degree-obvious is not the same as twelve-year-old-obvious. Describing points moving around another point is one of those ideas students can find hard to understand. I think it’s much better to let them create it for themselves.

 

“Why didn’t you show us that first?”

The gradient of ax + by = c is -a/b.

This is what Year 10 wish I had told them lessons ago.

They’ve been looking at parallel and perpendicular lines lately, which involves finding lots and lots of gradients. They like it when the equation of a line is in gradient-intercept form. The gradient is m from y = mx + b, and everyone is happy.

As an aside, why is m used for gradient/slope? Does anyone know? I’ve had kids ask me that so many times, and my honest answer has been “I haven’t got a clue.”

I was annoyed at the lack of images in this post. So here’s some parallel and perpendicular lines, just because.

Anyway, they aren’t so happy with standard form. They know how to rearrange between the forms, so they’ve been changing them into gradient-intercept form to find the gradient. But they’ve been getting really fed up with it. “This takes too long, Mr Carter! There has to be a quicker way.”

As I mentioned at the start of this post, there is a quicker way. But if I’d just given them that shortcut from the start, they would’ve been able to find gradients a whole lot faster. But they wouldn’t have understood why.

So I hinted that there is a quicker method. I wrote an equation on the board (I don’t remember what, I didn’t save my notes for some silly reason), and asked them to tell what to do to find the gradient. Which they did.

“So what’s the gradient of this?” I asked as I wrote ax + by = c on the board. There was a moment of quiet, before someone called out, “minus a over b!” Someone else added their agreement. Someone else asked “How did you get that?” Someone else disagreed, saying it can’t that simple. “Where did the minus come from?” asked another. And though they wanted me to confirm the answer, I just waited quietly for the class to come to their own agreement about what they’d just discovered.

They weren’t all that impressed with me when they realised I’d known this rule all along…

I feel like I should point out that this is not how the majority of my classes go. This is just an example of one time when everything went really right, and the class responded the way I wanted. My main motivation for writing this post is to remind myself that this type of moment, where I bait the class into their own discussions which lead to their own discoveries, is what I want to happen in my classroom. All the time.

Something I’m realising more and more is that mathematical knowledge has much more value to students when they earn it themselves. Would it have been more efficient to give the class the rule straight away. Well, yes, if by efficient you really mean “students were able to get more questions done”. But I don’t think that’s what efficient learning in class is. Rather than tell my students one specific rule that than can make use of in one specific scenario (and probably completely forget in a week), I want to teach my students to think mathematically and discover mathematics for themselves.

 

Graphing using intercepts (including worksheet)

There was sport on today, so a large portion of my Year 7s were away. As a result, I had an exchange with some of the remaining students at lunch time that went a little like this:

Student 1: We’ve only got 6 students in our class today! [Stretching the truth a fair bit, it was more like 15.]

Me: Okay.

Student 1: So we don’t have to do maths, right?

Student 2: Or if we have to, you can only make us do easy maths.

Unfortunately for them, I had a different idea. Sometimes when a lot of kids are away, it’s necessary to not continue with normal plans so those kids don’t get left behind. However, I still think that having the students that are left with me for 50 minutes is still an opportunity for them to learn.

So today, we looked at intercepts of linear graphs. This doesn’t really arise as a topic until Year 9 under the Australian Curriculum, but given that we’ve been doing a lot of work on solving equations and plotting graphs lately, it seemed like a good idea.

I’m a fan of giving students opportunities to see some of the maths they’ll use in future years, as long as it’s presented in a way that’s not too overwhelming. It allows them to see why we do some of the work we’re doing now, and gives stronger students a chance to achieve at that higher level. And it’s nice to be able to tell parents that their Year 7 child has been working on Year 9 work. 🙂

(I also may have told the Year 10s that they should get working, because the Year 7s were catching up to them…)

This was the process I came up with to let students discover intercepts:

  1. Define the x-intercept and y-intercept.
  2. Use Desmos to find intercepts.
  3. Notice the pattern in all the coordinates (i.e. there’s a lot of zeroes).
  4. Have students find x- and y-intercepts for equations using algebra.
  5. Sketch graphs for those equations. (This is their first time sketching, because it’s the first time using intercepts.)
  6. Use Desmos to check their graphs.

I created this worksheet to for the lesson:

The sheet is purposely vague in places, because I wanted students to figure things out for themselves as much as possible, and where they couldn’t, I could tailor my assistance as needed. Or preferably, they could help each other.

This task seemed to differentiate well, too. Some students needed very clear instructions on how to enter the equations into Desmos and how to read off the intercepts, and had to be reminded which axis was which. But once they did, they found the task fairly easy to follow.

On the other hand, some students recognised what was going on very quickly. I even had one student say, “Won’t all of them have a coordinate with zero?” before she’d even started the task. For these students, the process of finding the intercepts with Desmos was a way to verify the results they predicted.

You can download the worksheet here:

 

Football scores problem solution

The other day I posted this problem that one of my students discovered. We played around with it for a while and came up with this solution. I don’t know if this is the easiest or most elegant solution, but it’s what I have.

Quick recap, we’re trying to solve the equation 6a + b = ab, where a and b are non-negative integers. In Australian rules football, if a is the number of goals (worth 6 points each) and b is the number of behinds (worth 1 point each), the solutions to this equation are the scores where the total score is equal to the product of the goals and behinds.

Solutions can be found by trial and error, but how can we be sure we’ve found all of them? How do we know the solutions don’t just continue on forever? Well, it turns out that a little algebra and an understanding of asymptotes makes it clear that only a handful of solutions exist.

The original equation can be rearranged as so:

When graphed, this produces a hyperbola with asymptotes at a = 1 and b = 6 (see below). Which means that for a > 1, b is strictly decreasing and b > 6.

Since both a and b have to be integers, this puts an upper bound on a. Once b is 7, a can get no larger; if it did, b would not be an integer. As it turns out, b = 7 when a = 7.

We already have a lower bound: a = 0, as we only accept non-negative solutions.

So, we can simply use trial and error within this domain, and find these solutions:

Or, reported as football scores, 0.0.0, 2.12.14, 3.9.27, 4.8.32 and 7.7.49.

If we allow negative scores (not possible in football, but let’s do it anyway) we can get a few more solutions.

An interesting result is that b is always a multiple of a in these solutions. We actually noticed this result before finding all the solutions. a and a – 1 are coprime, so they share no prime factors. Since 6•a = (a – 1)•b, the prime factors of b must contain all the prime factors of a. A result that, as it turns out, didn’t end up helping us solve the problem, but is relevant to the name of this blog. 😉

The real question I have now is this: how do I go about turning this problem into a lesson?