Favorite secondary math manipulatives?

I read this is the mathematics educators stack exchange so hopefully this is the right place for this question. I was curious what is your favorite math toys, manipulatives, math games, or tools to use in your math class. Pretend your principal has said you could spend 1500 dollars and you teach secondary math.

• I use polydrons and snap cubes. See How can I demonstrate triangulations of surfaces with real hands-on objects? and snap cubes. – Joseph O'Rourke Nov 12 '19 at 10:38
• I presume you already have notebooks with squared paper, ruler, right angle, good 2B pencils, quality compasses, protractor and french curve. – Rusty Core Nov 12 '19 at 23:11
• Yes Rusty I have everything you said except the french curve. I honestly never heard of that tool. I have the polydrons. They love those. I don't have the snap cubes. – randomgirl Nov 13 '19 at 3:33
• Don't forget pentominoes ... – kcrisman Nov 13 '19 at 3:48

Secondary math is a pretty broad subject for something as individualized as the use of manipulatives to enhance teaching, but this is a list of things I'd love to have if I were teaching every subject in the same year for some reason.

• A two-pan balance scale and a bunch of clearly-labeled weights (Algebra I) - I feel that it can be really evocative how an equation like $$3x+4=16$$ can lead to $$3x=12$$ and from there to $$x=4$$ if you can manipulate labeled and unlabeled weights on a two-pan balance. The notion that doing the same thing to both sides to an equation is really brought home by seeing that that's what it takes for a balance to remain ... balanced.

• A wall-sized slide rule (Algebra II) - Logarithms had a clearly useful purpose fifty years ago that helped us to put humans in space. Slide rules bring that home. One of the sad stories of my teaching life is that my mother was at an estate sale and saw a giant promotional slide rule that was on sale for \$50. (Apparently, in the old days, if you bought a classroom full of handheld slide rules, you'd get a wall model for free.) My mom thought it looked cool but didn't buy it for me. Those things are usually$500, and I'd love to have one.

• A collection of 3D shapes (Geometry) - This one at least is affordable, and something I actually have in my room. They're a set of plexiglass polyhedra like a cube, rectangular prism, pyramid, cylinder, cone, hemisphere, and so on. There is a little hole in each one and they tend to share radii and heights when it would be convenient to use water to show that, for instance, the volume of a cone is $$\frac13$$ the volume of a cylinder with the same height and radius.

• A similar triangle "convincer" (Geometry) - I've never seen one of these before, but I'd really like my father to make a simple two piece jigsaw puzzle for me. (A 3-d printer would probably rock at this too. One of the key triangle similarity proofs is that a right triangle is split into two similar triangles by the altitude to the hypotenuse.

But I think it's hard for students to see because the common angle has to be "reflected". With manipulatives, it's easy to flip the piece over and see that it fits all of the angles and is the same size. I've never seen one of these for sale, but again I'd appreciate having it in my room.

I don't know if this is the right dollar amount, but I think getting access to a 3d printer and making some useful manipulatives of your choice with it would be cool. That is, if you are in a large district and have enough time to invest in trying them out - I don't think it's a one-off process!

As an example, Henry Segerman's website has some absolutely amazing examples - and the code is all available on Thingiverse, which is the go-to site for 3d printing.

And once you start seeing these, it's hard to stop looking for more! There are a lot of things out there, so you don't necessarily have to reinvent the wheel at all. Again, the caveats are important - this is a significant investment that is only worth it if you have people who would do it. But if you do, the sky's the limit.

• — do you have a link to that last article that isn’t behind a paywall? – Nick C Nov 12 '19 at 4:10
• I bought a 3D-printed sphere which makes a square grid shadow if you put a pen light at its north pole. It is the Stereographic projection which illustrates the conformality. I forget where I bought it, it was about $15, well worth it. – James S. Cook Nov 12 '19 at 5:08 • @NickC I'm very sorry that I do not. It should be accessible to Math Association of America members, if you know one. – kcrisman Nov 12 '19 at 11:22 • @JamesS.Cook yes, I saw a talk Segerman gave that showed off a variety of such projections, they are really cool. I don't have any classes I could use them in, but if I did ... ! – kcrisman Nov 12 '19 at 11:22 • Simple objects like the wireframe cube can — and I argue, should — be made by students themselves, because paying$11 is extortion and because they will get useful skills of handwork and learn something new about stereometry. More complex objects, well, how useful they can be? "Finding deeper meaning... touching... caressing?" — yeah, they are pretty, still not convinced in the necessity to drop big money on 3D printer. This is a fad, which pushes students further from hands-on learning. Instead, students should take drafting classes. "I don't have any classes I could use them in" — Q.E.D. – Rusty Core Nov 12 '19 at 23:22

I might think about some iconic objects of the past that excite interest, perhaps mainly in geometry. Drafting table (probably \$500 for a professional one, but you can check second hand). French curve as Rust mentioned. Slide rules. Parallel rulers and 10 point dividers (good for poking people when bored...I would know). Sextant. Abacus. Towers of Hanoi puzzle.

Different types of scientific graph paper along the walls or in framed pictures (and have a file of sheets as well): Regular, thick/thin for, semilog (and with decibel markings), log-log, triangular phase diagram, hex sheets, circular maneuvering boards. Posters with different geographic projections shown and explained. 2-D space group Escher tile posters. Random tables, used as wallpaper (log, trig, Bessel, etc.). Maybe some posters showing different functions/relations (doesn't need to be stuff they are learning...this is more for art and for intrigue): Bessel, epicycle, etc.

Handleable models of platonic solids and sphere packing (AB, ABC) and point group examples.

Get stuff that is sturdy and old and looks like it has seen the blood and sweat of combat. Part of the charm. Dress the place up. Make it look like a Hogwarts classroom. All sorts of mysterious instruments of mathematical torture arrayed on the walls and shelves.

As a no-cost option that gives you so many capabilities besides the conventional black/white-board, I always use geogebra. Not only as a demonstration tool, but also as a functionality for the students to investigate almost anything in precalculus or algebra.

By simply adding, e.g. some moving graphics you can easily explain why, say, $$e^x\approx\left(1+\frac{x}{n}\right)^n$$ for large values of $$n$$ or several other interesting but difficult to formally approach in secondary school mathematics.

• Although in general I think Geogebra (and other related options) are good, I don't think this answers the question about manipulatives. I do see that the question mentions "tools" but probably that is a bit too broad ... – kcrisman Nov 13 '19 at 15:38
• @kcrisman I can see your concerns. Thanks for your feedback. – Βασίλης Μάρκος Nov 13 '19 at 21:03
• Also, to the downvoter, may you explain your downvote so as to improve the answer - if considered necessary? – Βασίλης Μάρκος Nov 13 '19 at 21:03
• Oh, that was me! That was my concern - it doesn't mean others shouldn't up vote it, just that for myself, I think it isn't answering the question as posed. It is not a personal slight at all, I wish people would explain their down votes of my stuff sometimes, I can live with them :) – kcrisman Nov 13 '19 at 21:35
• Thanks! I know, whenever it happens that I downvote an answer/question I am used to post a comment below explaining why! :) – Βασίλης Μάρκος Nov 14 '19 at 9:43