Reading time: 10 min

Let me introduce Carl. Carl loves the holidays… and especially decorating the tree!

After all, this decision has a major impact, since the tree will be on display to everyone for nearly a month! Not counting its appearances in photos shared on social media (that’s another topic ;)). Okay. How can we help Carl choose his tree?

First, we’ll want to **compare the trees to one another**. And to do that, we have to find a model that can simplify our thought process a bit. Overall, this model should make the situation simpler, but still be realistic! So I suggest we talk about **TRIANGULAR** trees. If you’re not already convinced by my choice, maybe a quick example would be more effective. 😉

Like we said, to make a choice, Carl should **compare **the trees to one another! He makes the great decision to consider the height and width of the trees.

For example, when Carl observes this tree (let’s call it **tree 1**), he notes that the height is 100 cm and the width is 200 cm.

But Carl quickly realizes that it’s not so simple to rely on 2 values (height and width) for each tree.

He needs to find a way to simplify the information.

**determine the ratio between the height and the width**. To use the same example, Carl makes this calculation: (100 cm)/(200 cm) = 0.5. That’s the new label that he gives **tree 1**.

Carl gets to work and categorizes all the trees he can see! He quickly realizes 2 things:

- Several trees that are different sizes can have the same label (ratio).
- Some labels are ALWAYS associated with trees that are less than ideal.

Let’s consider these 2 discoveries one at a time!

Let’s go back to the case of **tree 1**. Its label is *0.5* because that’s the ratio between 100 cm (height) and 200 cm (width). But when you think about it, we could imagine a tiny little tree that is 10 cm tall and to respect the ratio of *0.5*, its width would have to be 20 cm. This tree would also have a label of *0.5*.

So I can conclude that this label could describe an infinite number of trees! In a way, we’re creating ** families of similar trees**!

What remains unchanged is the look of the trees in the same family. Indeed, there will inevitably be families made up of not-so-beautiful trees 🙁

Like Carl noticed, we now know that each *family of similar trees *contains an infinite number of trees. And since the *similar trees *have the same look, the comparison between trees boils down to comparing the families!

I’m wondering if there are some proportions (labels) that are more promising than others! In your opinion, what ratio is associated with the most beautiful tree? In other words, on a number line, where would you place the “Beautiful trees” label?

*divine proportion*, can be found everywhere in nature and according to certain sources, this number is associated with beauty. For example,

What I mean is that we can see a difference, but I think it’s difficult to say whether a ratio of 1.6 really is prettier than a ratio of 1.5 or even 1.4. In other words, the golden ratio may just be an excellent representation of all of the ratios whose proportions are pleasing to look at. It might be worth spending some more time on this idea in another episode 😉

It might be helpful to visualize this situation in the form of a graph. Indeed, here, each *family of similar trees *is found on the same line. The stocky ones, the slender ones, the well-balanced ones… no matter what the scale, all of the trees where the division of the height by the width is the same are aligned.

Finally, Carl has to make a decision! First, he identifies the areas of the graph that correspond to trees that are too *slender *or too *stocky*. Continuing the elimination process, he manages to determine his *favourite family*: trees that have a ratio between 1.5 and 1.7 between the height and the width of the base.

Carl just considerably reduced the range of options he has to choose from! But all the same, there is still an infinite number of trees!

And there you go! Carl finally has everything he needs to find the perfect tree and have the best holiday season ever…

.

Related activities:

- S2 Find the missing measurements in situations involving congruent or similar shapes: activity 1.
- S2 Solve geometry problems using algebraic expressions: activity 2.
- S2 Use proportional reasoning: activity 3 or activity 4.
- Mission that demonstrates the connection between the golden ratio and the Fibonacci sequence: activity 5.

**Disruption always works at a high velocity. Its impact may take some time, but the generation of new ideas, iteration, and dissemination into society are always moving fast. Just look at what Artificial Intelligence has done in the last few years.**

In 2016, the Chinese game of GO had one of the top players in the world quickly succumb to Google’s *AlphaGo *machine. What was astounding was not the actual defeat, but the strategies employed by the AI machine that ran counter to thousands of years of traditional strategies. *AlphaGo* basically made moves that were unconventional, random, and erratic.

Last year, the AI machine *Libratus* defeated some of the world’s top poker players in the world. Poker with its complex strategies, fragmented information, and use of psychology, was always thought to be almost immune from any technological challenges to this human game.

While these kind of developments are on the cutting edge of technology, every kind of technology–even digital math platforms–feed off of this massive disruption to conventional thinking.

In math education, this innovative speed becomes a complex game of business survival and meeting the high level pedagogical needs of math educators. Ironically, but not surprisingly, changes in math education occur much slower. This is for many reasons, which include, but are not limited to: tradition, historical need for formal education, size, and energy diffused in math debates on teaching practice/philosophy.

Technology also knows its place in the hierarchy of things. Well, *good *technology does.

So, not only does taking technology out of a supportive/reactive role to good pedagogy create a problem, but over-emphasizing its value negates the idea of equity in math education. It’s a tricky balance. However, the companies that understand the picture above will be the ones that thrive. The ones that don’t will be footnotes in the clogged highways of digital platforms.

The larger pedagogical ideas of mathematics, moving forward, are circling around play, sandbox learning, and deep understanding. Digital companies that are acutely aware of this are producing–*will need to produce*–resources that support the seamless confluence of these ideas.

A while back I created a series of short videos looking at the interactive nature of our platform. Here is a link to the popular *Modeling Multiplying Fractions* activity.

This constant pressure to evolve will not only help digital platforms become better in terms of meeting the educational students and teachers, but it will also exert influence on how mathematics has to change and adapt to the learning styles that are native to mathematics–ample space and time–*and* honor the interest/motivation of classroom students and teachers.

But, at the end of the day, teaching mathematics is about connecting with kids in ways that are genuine, human and memorable.

Technology can be a vessel or a barrier for that.

The students find themselves outside the Library of Alexandria, where the Greek mathematician Eratosthenes is trying to measure a globe. They are seeking a measuring tape, a device with units of measurement on it. With this simple instrument, the world of mathematics can take another giant step and Eratosthenes could perhaps measure the entire world!

To complete this mission, students will have to calculate perimeters, areas, circumferences, and distances.

Your Netmath sign in page has had a new look for a few weeks now. The design and the animation of everyone’s favourite Alfred set the tone for the changes planned for the platform. Now, you have the option to display the password that you enter to avoid any errors and connect more quickly.

Yes, there, right in front of you, on the home page, and only for your groups! We’ve revised how your students’ activation codes are made available: now they are displayed on the first page so that you’ll never have to search for them again. This should also help prevent errors (student activated by a teacher’s code and vice versa). Follow the steps on the screen and set sail on the Netmath adventure with your class in no time!

Connected, activated, and ready to play! Have a great school year on Netmath.