# Tag Info

2

If 10-11 year olds can learn programming (which some have done even before Scratch was a thing), then it's hardly a leap at all to suggest that 15 year olds can learn functions since they are a common element of programming languages. Every student is going to be different and some are going to be better at math than others, but I think this notion that ...

5

In all of these examples, I would draw a line unless that idealization led to missing something essential in the problem. These idealizations happen all the time, and they tend to be very useful. E.g., for the flow of water, we normally don't need to consider that water consists of discrete molecules. Ditto for examples like immigration, fluctuations in the ...

-1

I would do points, rather than a connected curve for A. Except if you are looking at large amounts, I wouldn't be so fussy and just draw a line. This actually has some use if you consider say different rates sold by a monopolist practicing price descrimination, for instance. (I think this is beyond your audience though.) For B, if you are interested in ...

3

Some abstraction is perfect for children. Mitsumasa Anno was a Japanese writer of children's books and he had many innovative approaches to introduction functions. This is from Anno's Math Games II. You can even see the introduction of function inverse.

11

Functions are far broader and more applicable than you give them credit for. Consider the following: Your students would be able to determine the elevation of Quito or the capital of Wyoming by interpreting that table far earlier than 15, and they're evaluating functions in order to do it. The only other thing you need to do is show them an example of ...

25

In the U.S. Common Core standards, functions are supposed to be introduced in the 8th grade, i.e., around age 13-14. So arguably age 15 is a year or two behind where they ought to be. The standard for the 8th grade says: Understand that a function is a rule that assigns to each input exactly one output. So honestly that really doesn't seem like a hugely ...

0

Good list. One of the above comments is true. In the modern classroom, if you wait to get everyone’s attention, you won’t be able to teach anything. That is especially true in today’s math classrooms. Many students are not motivated to learn. Math is objective and cumulative. It is easy to see where someone is on the spectrum of math understanding. However, ...

16

I think "function" is one of those notions that can be presented in different ways to people at different ages and who have different levels of ambition in math. This is similar to the notion of a "set," which I was taught in school ca. age 5 or 6 in an age-appropriate way, but then learned about at a different level in college. At a ...

9

OP: "Refuse to teach without attention." In my role as chair, I attended an instructor's class where he really refused to advance until he was certain the students were all with him, via detailed verbal feedback. The students responded, stopped the presentations and asked questions. I've changed my own teaching as a result of watching how this can ...

13

They are basic, friendly pieces of pedagogical advice. Most pre-college teaching is very much STILL in this mold. Where we fall down is in high-end universities and graduate schools, where pedagogy is less emphasized in the paradoxical belief that harder material should be learned with worse training methods. Or that smart students don't need/benefit from ...

3

I'd say yes, and I'd go with binary if you had to do any one alternative base simply because it's so relevant to computing and technology, and in my experience teaching discrete math, once you understand binary, related bases like octal and hex are pretty simple to pick up. But I don't think the converse is necessarily true. Ideally I'd like math and CS ...

3

I can't answer the OP's questions, but I'll just mention that a local 6th-grade teacher (in the U.S.) has a successful unit on base-$5$. It is mentioned in the recent article below. Sometimes he called it "star-fish math." James Henle. "Math for Grades 1 to 5 Should Be Art." Mathematical Intelligencer. 42, pages 64–69, Dec. 2020. ...

3

I really wish people would stop teaching the Pythagorean Theorem as $a^2 + b^2 = c^2$, for the following reason: Give your students the diagram below, and ask them to solve for $c$. At least 1/3 of a typical high school class will write $a^2 + b^2 = c^2$ and report back to you that $c = 5$. The problem is that the equation $a^2 + b^2 = c^2$ is so ...

3

Whatever you choose, make sure to follow some basic rules: Clearly state the preconditions and make sure they are understood (right triangle in your case). Clearly state the meaning of the symbols (e.g. which symbols stand for the sides adjacent to the right angle, and which for the third one). Use the symbols consistently (don't make e.g. the same symbol &...

8

Two alternatives I have seen used (am not necessarily recommending them, but will list some pros and cons) which don't seem to have been mentioned yet. Don't denote it algebraically at all! Draw a picture instead For the lower end of the 12-16 age range, I've seen this work really well. You literally draw the squares sticking out from the triangle. Write the ...

10

In Olympiad geometry, $a$, $b$, $c$ is the so-called standard notation for the sides of a triangle, so it makes sense to use it consistently when referring to a triangle (in isolation). However, in any case the general principle is introducing all your notation. Writing Pythagoras' theorem states that $a^2+b^2=c^2$. or Pythagoras' theorem states that $... -2 I prefer to state it the traditional way, not involving multiplying a length by itself, but involving areas of those polygons known as squares. Draw two lines at right angles to the hypotenuse at its two endpoints to get two sides of the square other than the hypotenuse itself, and draw the fourth side, and there you have a square, whose area is the sum of ... 19 Common knowledge The formula$a^2+b^2 = c^2$is common knowledge and the words for hypotenuse and leg (is "cathetus" not used in English?) are basic mathematical vocabulary. Including these seems a good idea. Connections to other mathematics The notation with AB, CA and BC might be something the students have used or will use in less analytical ... 6 The only one of these that looks objectionable to me is the one that calls the hypotenuse$h$, since in a triangle the letter$h$usually refers to the triangle's height (which could be either one of the legs but could not be the hypotenuse). 4 In engineering and physics, in the English literature, angular frequency is oftentimes abbreviated in frequency. Sometimes this is explicitly stated at the beginning, but mostly it is given implicitly. However, the intended audience is expected to not make any confusion. The usage is certainly not recent, and I think I've seen it already about 40 years ago ... 8 These books are simply reflecting the longstanding and universal usage in physics and engineering, which is that these words can have either meaning, and any ambiguity is normally either resolved by context or unimportant. 1 So some great books on geometry and written for middle/high schoolers are those written by Kiselev and translated to English by Alexander Givental. The textbooks are "Planimetry" and "Stereometry" respectively. Maybe try taking a look in either of those books and see if there is something that suits you? Here's a link to the first. https:/... 3 I took an undergraduate course in "advanced planar geometry" in preparation for secondary teaching. It was from the text by Isaacs, "Geometry for College Students (Pure and Applied Undergraduate Texts)" and is available for purchase here. There are also less upstanding ways to obtain this text. It appears to cover all the topics you've ... 1 @MatthewDaly's mention of The Secrets of Triangles reminded me of the just released A Cornucopia of Quadrilaterals, which I've been reading. (Alsina, Claudi, and Roger B. Nelsen. Vol. 55. American Mathematical Soc., 2020.) For example, for a bicentric quadrilateral (cyclic and tangential) of side lengths$a,b,c,d$and angles$A,B,C,D\$: \begin{align*} a + c &...

4

In my (lack of) experience, the second class in Euclidean geometry is actually an undergraduate course that seems to be often called College Geometry. And, yeah, there are so many fascinating topics there that are accessible to gifted HS students that don't really measure up to other topics when it comes to career and college readiness, but they were the ...

2

I wonder if you'd enjoy an older textbook on Solid Geometry. (The only ones I've ever seen are quite old. Maybe there's newer.) But that may not address the particular topics you mentioned.

8

I recommend either Excursions in Geometry by Ogilvy or Geometry Revisited by Coxeter and Greitzer. Both are cheap too.

3

A quick Google search returned the following texts based on key words from your posting: rational circle, Euler line, geometry textbook. Based on skimming the contents and prefaces, I think they are good chance to match your desires: A Beautiful Journey through Olympiad Geometry, Stefan Lozanovski, 2016 https://www.olympiadgeometry.com/the-book.html ...

0

It is absolutely a valuable skill. HOWEVER, students should also learn to be aware of the difference between finding the true answer and approximations. Approximations involve errors no matter how small it is unless it is zero.

1

A common error I see when teaching function composition is students seeing it as multiplication. Many factors contribute to this, but examples with multiplication in them don't help: Q: Let f(x) = 2x and g(x) = x+1. What is f(g(x))? A: f(g(x)) = 2(x+1) Some students will see this and think that the answer somehow involves multiplying f(x) by g(x).

6

You asked a big question. Maybe some of these ideas are worth investigating more deeply. The two first explanations are completely generic, while the last two are more specific to mathematics; I would advise to not forget the power of the generic explanations. But I don't know to what extent these are good explanations or what other good ones there might be. ...

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