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The post Calculus in a Chair appeared first on Geometiles®.

]]>Mr. Mayor reckons the chair is much more comfortable than it looks; Igor (pictured sitting) certainly seems happy. Regardless of Igor’s comfort, the chair is a great illustration of what we mean by a double integral. Those of you who are not familiar with calculus–or who never felt comfortable with it– please read on! This is meant for you!

Integral calculus is an amazing machine that enables us to compute areas — or volumes, if we’re in 3 dimensions– of objects of various shapes. Let’s use this view of the chair as an example.

Suppose we want to find the area inside the purple curve below. We will ignore the rest of the chair for now.

The most natural way to get an approximation of this area would be to find the areas of the thin rectangles inside it (let us assume that there are no gaps between them, as there are in the above picture). The better approximation we need, the thinner the rectangles and the more of them. The miraculous part of calculus is that the seemingly endless process of increasing the number of rectangles and decreasing their thickness is actually not endless at all. Assuming the curve is not too wacky, the process results in the true area under the curve. And this is what is known as a **definite integral**.

Here’s where Mr. Mayor’s chair is uniquely positioned to help. Suppose we want to find the volume of the chair. Again, let’s assume that the rods are right up next to each other, with no gaps between them. So our task is to find the volume of every rod. That’s something we can easily do without calculus. To keep organized, we could start out with the volume of the first row of rods in Figure 3 (above), and continue, row by row, toward the back of the chair.

But what do we do if the chair is upholstered, and we want the exact volume under the pink upholstery?

Just as before, calculus gives us the machinery to find the answer by increasing the number of rods as we make them thinner and thinner. With this fancy machinery, we can still keep our row-by-row scheme (we can also choose rows that go from the front of the chair to the back). And this is what is known as a **double integral**!

*Special thanks to designer Julian Mayor for permission to use pictures of his design, and to 3D modeler Pratik Parija for drawing Igor sitting in the chair, as well as the “upholstery”.*

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The post Archimedes’ Earring appeared first on Geometiles®.

]]>If you look closely, you will see that the post is a **cuboctahedron**, which is one of the Archimedean solids. A cuboctahedron is roughly what you get if a cube and an octahedron get married. It has the features both of a cube (6 square faces) and a regular octahedron (8 equilateral triangular faces). The cuboctahedron is called an Archimedean solid because it was first discoved by Archimedes in Ancient Greece over 2000 years ago. It was then re-discovered during the Renaissance. Here is a beautiful drawing of the cuboctahedron by Leonardo Da Vinci from around 1500:

And here is an cuboctahedron made with Geometiles:

Would Archimedes (or any other mathematician) ever have imagined this shape being used as an earring POST?

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The post Platonic Solids: Antiquity To Now appeared first on Geometiles®.

]]>Shapes have fascinated humans since the beginning of time. The ** polyhedra**, or solid shapes with flat polygonal faces, is one of such shapes. One of the most appealing types of polyhedra is the

As with most things, there is a functional and practical side as well. The twenty faces are symmetry-equivalent to one another, so, when the die is rolled, it has an equal chance of landing on any of the faces. An * icosahedron* creates what is known in the gaming world as a “fair die”.

Let’s dig a little deeper into why the ** icosahedron** has such a high degree of symmetry. Each face is an equilateral triangle, and that there are five equilateral triangle faces meeting at every corner. That’s interesting! But, what if instead of a triangle, we use

Johannes Kepler published the Harmonices Mundi, in 1619 (the page above is an excerpt). It illustrates the * five Platonic Solids*. Kepler, however, did not discover the Platonic Solids. Plato’s works mention them around 360 B.C. Plato believed each of the Platonic solids was associated with a particular element: fire, water, air, and earth. Since it has twelve faces, the

Kepler was fascinated with the construction of these solids and details this in his manuscript. For example, below, check out his breakdown of the icosahedron:

Geometiles® allows us to experiment with the construction of these solids just as Kepler did! Below, we can see how we could reconstruct the icosahedron:

Below you can see a modern rendition of the icosahedron I put together with Geometiles®. There is a lot of flexibility here, as I had the option of making each equilateral triangle out of two smaller triangles. Because of this, I could actually create patterns within the classic figure.

Reference for this section: Polyhedra by Peter R. Cromwell, Cambridge University Press, Cambridge, UK 2004.

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The post Summer Math Loss – How to Fight It and Prepare for the Coming School Year appeared first on Geometiles®.

]]>But what happens to children’s brains during that time of relaxation and fun? It’s a fact of life – during the summer months, young students will lose a certain amount of the information they have learnt during the school year. The exact loss is approximately two months’ worth of reading and mathematical skills, referred to as ‘summer learning loss’ or ‘brain drain.’ The gap in consistent education caused by the summer break may be yearned for by the students – those long summer days with no school are what dreams are made of – but it has an undeniable impact on the abilities of the children to retain the crucial information they learnt in the classroom.

Math loss is the most significant, and it’s easy to see why. Children can still be stimulated intellectually whilst on holiday: reading, museums, trips to the zoo, and other cultural activities will continue to enhance and complement the skills and knowledge learnt at school. However, it’s unlikely that these trips will be contributing to their knowledge of fractions and long division. Math gets neglected, not because it doesn’t exist, but because it’s harder to think of math as existing outside of the classroom.

So what can be done to counteract this?

There are, in fact, lots of ways to create and continue math-learning situations outside of school. They don’t have to be dull, or less engaging than other activities – they can easily be incorporated into the fun experiences that kids have while on holiday:

- Get your kids cooking, and incorporate math into the measuring process
- Point out the math in other activities – watching sport on TV? Talk about players’ statistics and results
- Take the kids shopping and have them help you when you work out change or discounts. It’s a whole new skill to be working out calculations in your head while handling bills, coins and interacting with people—compared to classroom math. See our Farmers Market Blog for details.
- Have a look at the math workbooks from their previous grade and check out what’s coming up when they return to school – giving them a head start will make it much easier when the summer is over.
- Get carried away with your set of Geometiles. Download our free online workbooks for structured learning, or just have kids explore shapes through free play. Visit us on Instagram to see some examples on how you can use it!

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The post Lemon Zest and the Surface Area of a Sphere appeared first on Geometiles®.

]]>Was I right? Just out of curiosity, I asked myself:

**What would be the size of the hypothetical one lemon which would yield the same amount of zest as my two small lemons?**

Would this be a gargantuan lemon of mythical size, or just a regular looking fruit? This was easy enough to determine, since my two small lemons were nearly spherical, with diameter of about 2 inches:

The lemon zest layer is so thin, that its amount can be measured by the surface area of the lemon times its thickness.

Using the formula for the surface area of a sphere of radius *r*,

Surface Area =4π*r²,*

and the fact that our lemons have a radius of about an inch, we have that t**he surface area of each lemon is about 4π square inches**. So the surface area of the two small lemons is 8π square inches*. *How large a single lemon would have to be in order to have surface area of 8π square inches? Its radius, *R*, is given by this equation:

8π = 4π*R²*

So *R*=√2≈1.4 inches. That’s not that big! At just 2.8 inches diameter, this hypothetical lemon would be well inside the range of lemons supplied by Sunkist growers. So it was a good call to replace this lemon with two little ones!

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The post Is Geometry the Unsung Hero of Mathematics Instruction? appeared first on Geometiles®.

]]>There is a widening gap between the effective teaching of geometry in elementary schools and the geometry skills students need in high school. Often, geometry tasks at the younger grades are limited to identifying shapes or labeling properties; in high school, students are expected to use abstract reasoning to prove a complex relationship. Dynamic, engaging instruction in geometry has traditionally been overlooked during middle school, causing a critical gap between elementary school experiences and the thought processes required in high school.

Geometry is all around us. It is part of our daily lives, whether we are at a cafe (see picture below), a construction site, or at the post office. Geometry gives us the tools to engage analytically with our everyday surroundings. It turns out that these tools provide us with more than just an amusing intellectual exercise. A growing body of research links spatial reasoning with future success in other academic pursuits. Interestingly, these pursuits reach far beyond geometry or even mathematics. Elementary school teachers and researchers at the University of Toronto found that students given lots of spatial reasoning exercises ended up doing better in numeracy, patterning and other areas of mathematics. In an unrelated study, researchers showed that spatial thinking skills are strongly related to students’ future success in STEM disciplines.

A common notion held by many people is that math is mostly about numbers.

This erroneous idea is part of the reason that many people write themselves off as being “not good in math”. Not only are geometry and geometrical thinking crucial components of mathematics, but they may also provide an excellent entry point to the subject for those students who think they are not interested or not good at math. Geometry is intimately connected with the visual arts—in fact, many leading artists of the Renaissance, such as da Vinci, Durer, and others took a keen interest in mathematics. Therefore, starting with geometry may spark an interest in math in students that would not consider themselves mathematically inclined.

In addition, geometry is, arguably, the area of math that lends itself most to being taught with hands-on tools, or “manipulatives.” These include pattern blocks, construction tools, or just sometimes just shapes that students cut out of paper. Manipulatives give students an opportunity to experience math kinesthetically, and this provides yet another entry point for those students who may be less receptive to traditional instruction. Since numerical and algebraic relationships are closely interwoven with geometric shapes, geometry provides a natural segue to other areas of mathematics. If you can get kids hooked on the importance of geometry before they come to the even more complex world of algebra and trigonometry, it will increase their chances of understanding and even loving mathematics later in their student and professional careers.

Geometry teaching in the US tends to be weak, even in high-achieving schools. Starting at the elementary school level, geometry is often relegated to the end of the traditional school year. By that time, both students and teachers tend to be tired, and the mental energy and time spent on geometry is often insufficient to do justice to the subject. Moreover, leaving geometry to the end robs students of the opportunity to see the essential connections of geometry to other areas of STEM throughout the school year.

It’s no wonder that, when students hit high school, they are often bewildered by the level of geometry at which they are expected to be working. If it hasn’t been introduced properly at elementary or middle-school level, they become very quickly disheartened by the struggle and may even give up the idea of math completely.

Instead of treating geometry as the crucial (and fun) foundational subject that it is, it is presented as a chore, and as the ‘down-side’ to math.

This must be addressed: if we enhance students’ geometry skills, we enhance their spatial intelligence and overall mathematical intellect.

Spatial ability or visualization is essential in engineering and scientific fields. It is a skill that is required for the effective generating and transforming of visual images. Verbal IQ tests and Scholastic Aptitude Tests (SATs) are great for measuring the two other critical cognitive abilities- quantitative and verbal- but they do not sufficiently measure spatial ability, and those who are gifted in the area often go unnoticed.

Thus, if we neglect geometry (and therefore spatial ability) in the school curriculum, as well as in standard assessments, the students with spatial strengths will be in the minority- and it is they who are required to push the boundaries of the technical and scientific professions.

For more, see:

- 7 Ways to Boost your K-8 Math Program
- Mathematical Discourse: A Matter of Sharing and Questioning
- Middle Grade Math: Comprehensive and Open

*This article was first published on Getting Smart.*

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The post Geometiles at the Julia Robinson Mathematics Festival appeared first on Geometiles®.

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The post 3 common classroom challenges and suggestions for handling them appeared first on Geometiles®.

]]>- A classroom full of students at different levels, despite the fact that most students in the math club self-selected to be there
- Creating an atmosphere where students feel emotionally safe enough to participate in the class– meaning to risk providing wrong answers
- Assessing your students’ understanding, other than by testing

The three challenges are interrelated, and I will describe some ways of addressing them. Most of the ideas are gathered from other successful teachers and were chosen because the issues they address echoed my experiences in Math Club.

This is obviously a loaded subject, and I’d like to focus just on the interpersonal dynamics of a class in which students are performing at different levels. Let’s be honest: math clubs tend to attract high performing students, and, in my experience, such students often have a strong desire to demonstrate to everyone else just how high performing they are. If you are starting a math club, you need to be attuned to this from the very beginning. Otherwise, some serious damage can be done to the learning experience of your students who are either not as high performing, or less confident. My personal suggestion is to nip in the bud any behavior that smacks of arrogance. Do not wait until the damage is done.

If there is one phrase that I could eliminate from any math club discussion, it would be “This is easy”. This phrase, and its consequences, are addressed in an article by Tracy Zager, a well-known math teacher and author: “This is easy: The litte phrase that causes big problems.” Zager explains the title by describing the paralyzing effect this phrase has on students who are struggling with a problem and hear one of their classmates make this announcement.

All teachers know how much more fun and engaging a class can be if students participate. But this comes with a risk: what if the answer a student provides is wrong? This is beautifully addressed in the blog post 5 Ways to Respond When Students Offer Incorrect Answers by Harvey Mudd College Professor Rachel Levy. Although the intended audience of the article are college professors, it applies equally well to teachers and coaches of students of any age. The only thing I would add to Professor Levy’s ideas that is particularly suitable for a younger students’ math club is to keep the atmosphere light, even goofy. Make jokes. Make fun of yourself. One extremely successful teacher, Anthony Yom, even let his East LA students make fun of him for knowing nothing about authentic Mexican food.

Guilty as charged: I tend to say this a lot. But a recent article by teacher Dylan Kane made me realize that this may not be the best way to assess my students understanding. In his blog post Things Not to Say, Kane says that this question may not be as innocent as it sounds. “… the implication here is ‘whoever feels confused, please share your inadequacy with the class’. Instead, I can ask students what was unclear, or ask a more specific question that acts as more productive formative assessment.” I would add to this that sometimes, an anonymous online survey (for example, by Google) can give you lots of data about the holes in students’ understanding that you wouldn’t otherwise get in class.

Be patient with yourself if you are struggling to create a classroom environment that you would like to have. It’s often not an easy, one step process. Talk to your fellow coaches. Find a family you’re comfortable with and talk to the child and/or the parents to gather their feedback about the classroom environment. Most importantly, learn from your and other coaches’ mistakes.

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The post Platonic Solids Poster appeared first on Geometiles®.

]]>Here are some of our blogs about Platonic and Archimedean solids:

- Public art installation in Los Angeles featuring huge Platonic and Archimedean Solids
- Examples of Platonic Solids in antiquity

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