Virtual pattern blocks are not hard to find on the Web. One good implementation is on the Math Learning Center site. Among other valuable features, it allows the selection of a group of blocks which you can copy-paste, for example in creating a pattern block tiling. One inexplicable choice was the inclusion of a purple isosceles right triangle. This makes no sense to me because its hypotenuse will not match the sides of any of the actual pattern blocks. (On the other hand, one good use of this purple block would be to discover the supertangrams, but there’s no reason to include it in the same applet as the pattern blocks.)

I was interested in creating more specialized pattern block applets, which would focus on specific activities. My first creation along these lines is to support Lab 5.6 from *Geometry Labs.* The lab is an exploration of symmetry, in which students cover three figures with pattern blocks. I started with Jen Silverman’s GeoGebra pattern blocks, and added the figures. This is what it looks like:

The beauty of virtual manipulatives in GeoGebra is that you have access to the full power of the software. For example, I used the rotation tool to make this figure:

And this one, using reflection across just two mirrors, executed repeatedly:

(Look at more double-sized pattern block dodecagons on my website.)

However, the limitation of GeoGebra for virtual manipulatives is that it’s inconvenient and complicated to create multiple copies of blocks, which is of course necessary for pattern blocks. The existing GeoGebra Rigid Polygon tool has the potential to solve this problem. The tool duplicates any existing polygon by simply clicking on it, and the copy can be rotated “manually” by dragging one of the vertices around another. This could be used to create as many copies of the original pattern block as you want! The problem at the time of this writing is that the copies do not inherit the properties of the original: the colors, styles, and scripts are lost. If that was fixed, GeoGebra would be a terrific environment for virtual pattern blocks (and any virtual manipulatives used for tiling.)

Multiple copies is not an issue for tangrams, and I was able to create a decent GeoGebra Virtual Tangrams applet.

This applet will work for the tangram activities in *Geometry Labs, *or really any tangram activities, but it does not add anything to what you can do with more enjoyment using physical tangrams. I expect that a similar approach will work for other geometric puzzles, such as pentominoes and supertangrams. However those will take more work due to the greater number of pieces, so I have not yet embarked on those projects. [See my update below!]

There are a number of online geoboards, some of them beautiful. The best of the lot is the one on the Didax site, which makes it possible to do all the geoboard activities in *Geometry Labs*. It looks like a physical board, includes rubber bands in five colors, and unlike most of the others, it has the 11 by 11 dimensions needed for the *Geometry Labs*. And it even has a circle geoboard version. Still, I decided to make my own GeoGebra Virtual Geoboard and Virtual Circle Geoboard.

They do not try to look like physical geoboards. Instead, they make it possible for a tablet user to handwrite or draw directly on the board, not unlike what one might do on a whiteboard, except that the figures are extremely accurate. Here are two examples:

Also note that the inside of polygons can be colored, a possibly useful feature that is not available on most online geoboards.

Along the same lines, I made some Virtual Grid Paper — not exactly a manipulative, and not an electronic graphing utility, but useful in doing all sorts of math.

- The main purpose of manipulatives is to enhance classroom discourse. Virtual manipulatives are not as effective for that as the physical kind, and should not merely imitate those. Rather, they should take advantage of the power of the computer to enrich the experience.
- To avoid some common design mistakes, virtual manipulatives should be created in consultation with educators who understand how they might be used in the classroom. Or, they could be created by the educators themselves. This has become more possible thanks to platforms such as Google Drawings and GeoGebra. Each platform has its shortcomings, but they do allow an amateur like myself to create useful learning environments, closely tied to curricular materials. I link to all my virtual manipulatives
**here**. - Google Drawings allow the manipulation of virtual objects in a way that is vastly more accessible than traditional graphics software. Figures can be annotated with (typed) text. Multiple students can interact with the figures simultaneously, in a way that could promote collaboration.
- GeoGebra is of course full of tools of every sort, some of which can profitably be included in a virtual manipulatives applet. In this post, I mentioned the geometric transformation tools, the availability of “snapping” to a grid for accuracy, and the possibility of handwriting right on the figure.

Many online manipulatives are pale imitations of the real thing. Some are well-thought-out imitations of the real thing. And some are in some respects superior to the real thing. In this post, I tried to suggest some ways to move in the direction of the third category.

I used GeoGebra to make Virtual Pentominoes. Read on to see suggested sample lessons.

Adjust the rectangle to the desired size, and solve the puzzle:

What rectangles are possible?

Or, hide the rectangle and use the Polygon tool to create your own puzzle, for example by doubling or tripling pentomino dimensions:

(The two types of visible vertices are how you turn and flip the pieces.) Now solve the puzzle, and hide the vertices:

In this example, we doubled the dimensions. What happened to the area? What would happen if we tripled the dimensions? Which puzzles of this type can be solved?

]]>In 1981, after ten years in K-5, I switched to teaching high school math. In some ways, this felt like starting a whole new career: the math was more involved, the relationship with students less like parenting, and tradition weighed a lot more heavily on the profession. Still, in other ways, teaching is teaching, and much of what I learned in K-5 still applied. For example, “Michelle’s table is ready!” works a lot better than “Stop talking and listen to me!” Also: puzzles and learning tools, which had dramatically enhanced my math lessons in elementary school, could also be used to improve high school classes.

Yet high school teachers didn’t use manipulatives much, if at all. The dominant idea seemed to be that clear explanations followed by silent practice on paper was the gold standard. I knew from experience that this was not a particularly effective approach on its own, and started to complement it with traditional elementary school materials such as pattern blocks, geoboards, tangrams, and pentominoes. I was thrilled at how well that worked. Encouraged by this early success, I designed new manipulatives for use in math classes: the Lab Gear (for algebra), a well-thought-out geometry drawing template, the CircleTrig Geoboard, (both for *Geometry Labs,) *and supertangrams. I also had the amazing privilege to work with George Hart on the *Zome Geometry* book.

(Find links about manipulatives in middle and high school here.)

This interest in manipulatives does not stem from a naive belief that if students can touch it, they will understand it. (“Magical hopes”, as Deborah Ball put it back in the day.) The usefulness of manipulatives depends on how they are used: as with any approach, reflection, discussion, connections, and generalization are crucial. But that’s the point: with proper teacher leadership, manipulatives can offer a concrete environment that increases student interest, levels the status playing field, facilitates communication, and stimulates mathematical thinking.

Likewise, rich computer learning environments provide opportunities to raise the level of intellectual engagement — as long as they are used well. (Alas, it is easy to misuse electronic tools… See my article on this topic.) By “rich environments” I mean electronic platforms that support teacher and student creativity: Logo and its descendants such as Scratch and Snap, electronic graphing, interactive geometry, and the like. True, screens can be hypnotizing and isolating, but the Desmos team has found ways to make their graphing software part of lively classroom discussion, and it seems like GeoGebra is trying to catch up on that front.

(Over the years, I’ve written a lot about learning tools. See especially For a Tool-Rich Pedagogy which features both a philosophical argument and dozens of relevant links.)

Virtual manipulatives attempt to bridge the gap between high-tech and low-tech learning tools. That attempt has not always been successful. In some cases, the tools were just poorly designed, perhaps because the designer didn’t understand how the manipulatives are intended to be used. (Two recent examples: tangrams whose parallelogram cannot be flipped; pattern blocks that rotate in 5° increments, so that it takes six clicks for the smallest useful rotation.) In other cases, the design was decent, but did not use the power of the computer, so that switching to an electronic version of the manipulatives led to a loss in classroom interaction, with nothing gained.

However, in a time of remote instruction, virtual manipulatives cannot be compared with physical manipulatives, since those are typically not available in students’ homes. The comparison is with not using manipulatives at all. I don’t have a lot of experience teaching remotely, but I’m pretty sure that virtual manipulatives are better than no manipulatives. As is my wont, after a little bit of research on what’s “out there”, I started making my own. This is now a lot easier than it would have been even a few years ago, as there are graphical and computational environments that put this within the reach of an amateur like myself. I am thinking of Google online apps, and GeoGebra, but I’d love to hear about more (preferably free) options.

I will use this post and the next to think about virtual manipulatives, and to let you know about my attempts along these lines.

Some years ago, I used GeoGebra to create applets to explore and discuss multiplying binomials and squaring a binomial, using the Lab Gear model. What I liked about the applets is that if a teacher could project their laptop screen, they had a way to discuss these ideas with a whole class. The applets could also be used by students, for example to create figures and illustrate a report. Another applet, completing the square included specific questions that students can answer to show and consolidate their understanding of an algorithm that pre-manipulatives was exceedingly difficult to learn.

That’s all well and good, but these uses only make sense after the students have had plenty of experience with the physical blocks. And moreover, these applets each had one single purpose. The physical manipulatives can be used in a huge range of algebra topics: signed number arithmetic, polynomial factoring, equation solving, and so on (as you can see in the Lab Gear books). Clearly, useful as they are, these applets do not qualify as virtual manipulatives.

To address that, and to respond to a teacher’s request, I used Google Drawings to make Virtual Lab Gear. What’s great about those is that several students can work together in one drawing, and one person (teacher or student) can demonstrate something for all to see. In addition to live manipulation, the drawings allow for the creation of Lab Gear illustrations by anyone — a long-standing request of Lab Gear users. In turn, those illustrations can be copy-pasted into other applications.

I was pretty sure that this Virtual Lab Gear would make it possible to carry out all the activities in the Lab Gear books. However, no sooner had I posted the drawings that a correspondent pointed out that it was not possible to use them for “Face to Face” (Activity 1-1D in the *Algebra Lab Gear: Algebra 1* book). This is an activity I like, because it previews a key idea: common dimensions correspond to common factors. It gets at that by asking students to place the blocks on rectangles that represent the various possible faces.

Here is an alternative which can be done using Virtual Lab Gear. The idea would be to make “trains” or “towers” of blocks. Each would consist of all different blocks. (No repeats within a train or tower.) The blocks would connect face-to-face, only on congruent faces. The assignment would be to do this for 1 by 1 faces, 1 by x, x by x, and so on.

Here is a 1 by 1 train I created using Virtual Lab Gear:

Here is an x by x tower:

and so on. I suspect that this is in fact an improvement over the original activity, whose instructions for some reason were hard to communicate to students. (I really did not expect that the virtual manipulatives would suggest an enhancement over the physical version!)

If I create such a worksheet to substitute for 1-1D, I’ll post it on the site. And if you do, send it to me! In fact, if you use the Virtual Lab Gear in creating PDFs, or slides, or whatever, I’d love to see what you did, and I might share it on my website. (Crediting you, of course.) Also, as always, I would appreciate any feedback on this.

Well, that’s all for now. To be continued! I’ll share some more thoughts and some more virtual manipulatives in my next post.

— Henri

]]>A few days ago, I saw videos by Jeffrey Smith: sum of cubes, difference of cubes. Those were the inspiration for this blog post.

When I designed the Lab Gear, I made sure it allowed for the representation of (x+y)^{3}:

But seeing Smith’s video made me realize the Lab Gear also allows for a hands-on representation of the sum and difference of cubes — albeit in a non-obvious way.

Let’s start with the sum x^{3} + y^{3}:

Let’s build on top of those blocks, in order to get two towers of height x + y*:*

And now, let’s write the total volume of the towers two ways. Block by block, we see that the volume is

x

^{3}+ y^{3}+ x^{2}y + xy^{2}= x^{3}+ y^{3}+ xy(x + y)

As two prisms, we see that the volume is

x

^{2}(x + y) + y^{2}(x + y)

Write an equation to show these expressions are equal, and solve for x^{3} + y^{3}:

x

^{3}+ y^{3}= x^{2}(x + y) + y^{2}(x + y) – xy(x + y)x

^{3}+ y^{3}= (x + y)(x^{2}– xy + y^{2})

Admittedly, following this argument requires some facility with algebraic manipulation, and I do not recommend it for students before Algebra 2 or precalculus. Still, as a teacher, I appreciate seeing a geometric interpretation of the formula. It is also interesting that the standard “Make a Box” Lab Gear activity is applied here in a slightly different version: “Make Two Boxes”. The key idea that makes this work is that the boxes have the same height. As always with the Lab Gear, a common dimension represents a common factor.

On to the difference of cubes! This one will require us to make three boxes, all with the same height. However, we must start by choosing new variables. We will use a instead of x+y, and b instead of x. Given that, this building represents a^{3} – b^{3}:

Take the blocks on the top layer, and rearrange them so we have three towers, all of them with height (a – b):

The volumes of the prisms are ab(a – b), b^{2}(a – b), and a^{2}(a – b) respectively. So:

a

^{3}– b^{3}= (a – b)(a^{2}+ ab + b^{2})

QED!

— Henri

]]>———————————————————————

A long time ago, in my twenties, I attended a lecture about the mathematics of wallpaper designs. The presenter gave an overview of the entire proof that there are only 17 different wallpaper groups. (*Groups* are mathematical structures which are foundational to symmetry, to the Rubik’s Cube, and to the number systems we use every day. Learn more about this in my Abstract Algebra unit.) I probably did not follow the whole argument then, and I certainly don’t remember it now. Still I have no trouble believing that from the point of view of symmetry, there are only 17 possibilities for an infinite repeating two-dimensional design (a wallpaper).

In this post, I will try to get across some reasons why possibilities are so constrained. In the last post, I analyzed wallpaper designs that include both six-fold rotations and mirror lines, and showed that several such designs have exactly the same structure. Today I will show that given that assumption, no other structure is even possible: if a repeating design includes a mirror and a six-fold rotation center, then it must be structurally identical to the designs we analyzed in the previous post.

The argument I will make will be visual and geometric, and even though it will not be 100% rigorous, it should be sufficient to convince you that the mathematical choices for a wallpaper design are quite limited.

We will need some basic facts to help us navigate this argument.

**Fact 1:** If two mirror lines intersect, their point of intersection is a center of rotational symmetry.

For example, if two mirror lines are perpendicular, then an item such as the shape labeled 1 in the figure is reflected as shape 2 in one mirror. Shape 2 is reflected as shape 3 in the other mirror. Shape 3 is reflected as shape 4. The overall result is 2-fold rotational symmetry, centered at the intersection of the lines.

If the lines make a 60° angle with each other, we can follow the reflections alternately in a similar way, leading to three-fold rotational symmetry:

If the lines make a 30° angle with each other, we get six-fold rotational symmetry:

We will encounter all three of these situations below.

(You can and should try this experiment yourself using GeoGebra or any interactive geometry software: start with two intersecting lines, and see what happens when you reflect an asymmetric shape repeatedly, alternating between the two lines. Depending on the angle between the lines, you will get more or less interesting results. What angles yield the most visually pleasing results? It is also worthwhile to carry out the same process with two parallel lines, or three lines that make an equilateral triangle, etc.)

**Fact 2**: The reflection of a mirror is a mirror.

This is because if a line is a mirror line, the whole design is reflected in it. Here is an example based on the above figure:

The new lines were all created by reflecting the original mirror lines in each other, and the resulting lines in each other, and so on. All of them are indeed mirrors. Those symmetries were already in the figure, a consequence of having the original two mirrors.

Similarly, the reflection of a center of *n*-fold rotation across a mirror line is a center of *n*-fold rotation. The image of a mirror in a rotation by the appropriate angle around a center of rotational symmetry is a mirror. The image of a center of *n*-fold rotational symmetry in a rotation by the appropriate angle around a center of rotational symmetry is a center of *n*-fold rotational symmetry. A concise way to put all this: *symmetries propagate each other*.

We are now ready to see why if we have six-fold rotational symmetry and a mirror line in a repeating design, then we must have the entire structure we explored in the previous post.

First, let us assume that our center of symmetry is *not* on our mirror line, like this:

Because symmetries propagate each other, there must be another six-fold center on the other side of the line:

For the same reason, there must be a mirror that is the result of rotating our original mirror 60° around that center:

That mirror goes through our original center. (This is not an accident. I won’t do it here, but it can be proved using basic high school geometry.) But since we have six-fold symmetry, rotating this new mirror 30° around our original center gives us a new mirror:

Our new mirror passes through our second center.

Let’s now show all the mirrors through our two points:

(I color-coded the points and the mirrors more or less with the colors I used in the previous post.)

But look at all these mirror intersections! As we saw above (Fact 1) these are all centers of rotational symmetry. If the angle is 90°, two-fold. If 60°, three-fold. If 30°, six-fold:

As you see, this is exactly the pattern we saw in the previous post. Perhaps it will be more obvious if I show a little bit of the hexagon tiling that got us started:

What I tried to show is that from the point of view of symmetry, there is only one way for a wallpaper pattern to have both six-fold rotational symmetry and mirror symmetry. Starting with a single mirror and center, plus some math facts, we found that the pattern had to include a whole network of mirrors and centers, including some two-fold and three-fold centers, all of them organized in the pattern known as **p6mm** by crystallographers.

(I started the argument with the original center *off* the original mirror. A very similar argument would work if our original center was *on* the original mirror.)

I hope I convinced you that the options for symmetry in two dimensions are severely constrained. Not everything goes!

For further exploration, visit the pattern blocks Wallpapers Catalog on my website.

— Henri

]]>In this post, I hope to explain what I meant. This is not going to be an exhaustive analysis of the wallpaper groups, only a clarification of what we mean when we say that mathematically, there are only 17 possibilities. (Of course from other points of view, the possibilities are infinite!) I will not discuss this from the point of view of abstract algebra: my approach will be visual and geometric.

We’ll start with an example. Many bathroom floors are tiled with regular hexagons, like this:

(In the rest of this discussion, you must imagine that the tiling extends infinitely in all directions.)

Let us analyze this design’s symmetries. First of all, imagine a line that joins opposite vertices of one of the hexagons, like this:

This is a line of reflection for the (infinite) tiling, a mirror line. Can you see how each hexagon, or half-hexagon is reflected on the opposite side? The same is true of all such lines:

(I did not draw all of them — again, you have to use your imagination to complete the picture.) Reflections in these mirror lines are among this tiling’s symmetries.

But there are more mirror lines: the ones we would get by joining the midpoints of opposite sides of the hexagons:

Reflections across those lines are also part of this tiling’s symmetries. The next figure shows both sets of mirror lines:

There are no other mirror lines, but there are more symmetries: rotation symmetries. A figure is rotationally symmetric if after turning it less than a full 360° around a center, it appears unchanged. Our hexagon tiling has three types of rotation symmetry: 2-fold (180°), 3-fold (120°), and 6-fold (60°). The figure shows the centers of symmetry, color-coded:

If you turn the whole plane 180° around one of the red centers, the figure will appear unchanged. Around the green centers, you can turn 120° or 240°. And around the blue centers, 60°, 120°, 180°, 240°, or 300°. These rotations are also among the tiling’s symmetries.

All of the symmetries are represented together in this figure:

Now let’s analyze this tiling:

The tiling is more complicated, but as you will see it has exactly the same symmetries:

- Mirrors between adjacent blue tiles
- Mirrors joining opposite points of the six-pointed stars
- 2-fold rotation centers at the common vertices of hexagons
- 3-fold rotation centers at the centers of the hexagons
- 6-fold rotation centers at the centers of the stars

Here is a figure showing all the symmetries:

And here are the lines and centers of symmetry, without the distraction of the tilings where we found those:

The second image is more spread out, but notice how they have the same structure. Each blue dot is surrounded by alternating red and green dots forming a hexagon. Each green dot is surrounded by alternating red and blue dots, forming a triangle. Each red dot is surrounded by alternating blue and green dot, forming a rhombus. Some lines go through red and green dots alternately, while others follow a green-red-green-blue pattern.

In other words, *from the point of view of symmetry*, the two tilings are equivalent, even though they are quite different on the surface. Here are two more pattern block tilings that share exactly the same structure. Can you find the mirror lines? the centers of rotation? Can you confirm that the mirrors and centers are arranged, again, in exactly the same way?

(The latter image was contributed by John Golden. Be careful: not all the hexagons have the same job in his tiling!)

Crystallographers call this symmetry pattern **p6mm**. It is one of the seventeen possible wallpaper patterns. From the point of view of symmetry, all four tilings above share the same structure: mirror lines and centers of rotation are organized in exactly the same way. This is not a coincidence! In my next post, using this pattern as an example, I’ll try to explain why wallpaper patterns are so constrained.

— Henri

PS: For the purpose of this post, I made my tilings using the Ontario Ministry of Education applet. (I added the mirror lines, the rotation centers, and the legend using Affinity Designer.) John made his tiling using the Math Learning Center applet.

]]>As far as I can tell, WordPress only shows you the animation once, when you open this post. You can find the actual **.ggb** file, and an interactive applet in your browser here. (Links to related transformations applets are on this page.)

The idea was to illustrate reflection in a line and help students get a feel for the transformation. Mathematically, this is ironic, since transformations of the plane in fact do not involve motion at all: they are functions relating two static figures. Technically, each point in the original figure is an input, and a single point in the image is the output. Nothing moves!

Still, the animation helps, and it does illustrate how we think about transformations intuitively. A translation “moves” an object. A rotation “turns” it. A reflection “flips” it. In fact, in the above animation there’s a bit of a 3D flipping illusion.

I am not a GeoGebra expert, far from it, but in this post I’ll explain my approach to creating **.ggb** animations. I’ll use the reflection as an example, but the same basic principle can be used for many simple animations. If you know a better way to do any of this, please let me know!

These instructions work in the current version of GeoGebra 5. I hope they would also work in other recent versions of the software.

(Note: the approach outlined below is different from and better than the one I used to create the applet cited above. This is the file I created while writing this post.)

As you work through this tutorial, you might learn some non-obvious GeoGebra features.

1. Make a **slider**, with **Min 0**, and **Max 1**. This will control the animation.

2. Make a **line**, which will serve as the reflection line. Make a **polygon**, which will be the pre-image. (When exploring transformations, it’s best to use an asymmetric figure. I usually make it orange, a reminder it’s the *original* figure.)

3. Use the **reflection** tool to make the reflected polygon. (I usually make it indigo, a reminder it is the *image*.)

4. Make **vectors** connecting each of the vertices of the original polygon to its image.

Here is the figure as it stands now:

(Digression: Notice that the vertices of the original polygon are fatter than the ones in the reflection. That is a setting in **Preferences → Defaults → Point → Style**, where I set the **Point Size** to be smaller for **Dependent** points. This makes it obvious which points can be manipulated directly, and which can’t.)

And here is the key idea: we will translate each original vertex by its own vector, multiplied by the slider number. When the slider is at 0, the point’s image stays in place. When the slider is at 1, the point’s image will have reached the end of the vector, in other words the original point’s reflection. When the slider is in between, the point is in between, traveling along its vector. Here’s how you do it:

5. Show the **Input Bar** (in the **View** menu.) Create a new point by typing:

**G = Translate(C, a*w)**

where **G** is the new point we are creating, **C** is a vertex of the original polygon, **a** is the slider, and **w** is the vector that starts at point **C**. (Obviously, you should use the appropriate letters in your figure.)

Now, if you slide the slider from 0 to 1, you’ll see your new point moving from the pre-image to the image.

Do this for all the vertices of the original polygon.

6. Make a new **polygon** by connecting the new points in the correct order. I made it grey:

Success! If you move the slider back and forth between 0 and 1, your new polygon moves from the original to the image, and back.

What remains to be done: making sure that the various parts of the figure are appropriately either visible or hidden.

7. The vectors and the grey and indigo polygons’ vertices need not be visible ever, so you should hide them.

8. The grey polygon itself should only be seen while on the way, not when the slider is at 0 or 1. This is accomplished by selecting the grey polygon, and going to **Edit → Object Properties → Advanced**. There you can enter a **Condition to Show Object. **Type:

**a > 0 && a < 1**

(My slider’s name is **a**. Use your slider’s name!) **&&** is how you enter the operator “and”, which is then shown as **∧**. You need to repeat this for all the grey polygon’s sides, which you can do by copy-pasting the condition in their own properties.

9. Finally, the indigo image should only appear when the slider is at 1. So its condition is

**a = 1**

which should be entered for the polygon and its sides.

You’re done! Check that it all works as expected.

I have used this basic scheme for many animations I’ve created in GeoGebra. The mathematical heart of the strategy is the scaling of the vectors using a common multiplier, which is controlled by the user. For this to work, you need to first create the final image, which is needed to make the vectors. Among other things, this approach will allow you to animate translations and dilations. (Rotation requires a somewhat different process, where it is the angle of rotation that is scaled by the slider multiplier.)

Pedagogical note: the reason I prefer to leave the control of the animation in the hands of the user is that it allows them to think about the figure at their own pace, and backtrack if needed, rather than passively watch a video. It also makes it easier for a teacher to lead a discussion of the figure, step by step, while projecting it on a screen. To see applications of this approach, check out all my GeoGebra applets. Many of them incorporate user-controlled animations.

— Henri

]]>- Catchphrases, where I mostly discuss the assorted slogans I have spouted over the years.
- How To, where I argue that there is no single “best way” to teach any particular topic.
- Eclectic, and the subsequent posts, where I challenge fads, but also say we can learn from them.

Of course, this does not mean I believe all “ways” are legitimate at all times or all ways are equally valid. For example, I’m a big user and promoter of learning tools. Shira Helft introduced me to the concept of *the knife*: a learning tool that is powerful and versatile, like a kitchen knife. (As opposed to specialized tools such as peelers, corers, slicers, pizza cutters, and so on.) Shira is right: teachers should prioritize multi-use learning tools, especially at the start of their career. Later, they can diversify, as I did. (I discuss this here.)

Still, I insist that it is a mistake to put all of one’s eggs in any one instructional basket. The main reason is pedagogy: no one tool, no one strategy, no one model works for all students, all classes, all teachers, all communities, every day. (See my cheerfully titled article: Nothing Works.) One’s growth as a teacher is largely about learning many ways to teach important concepts, so as to reach more students, and to deepen all students’ understanding. How many ways you can think of something is an excellent measure of how well you understand it.

**Frequently asked question**: doesn’t using multiple approaches confuse the students? Well, yes, it may. It is especially confusing when connections are not made between the approaches, or when the teacher’s understanding of an approach is superficial. But we run greater risks if using a single approach. Multiple representations (and thus multiple approaches) are dictated* by the math itself*. This is what I want to discuss in this post.

Here are some examples with whole numbers.

A good way to think about whole number subtraction is with the help of counters. If I start with ten counters, and remove three, how many are left? Another good way is to think of whole numbers as sitting on a number line. If I start on the 3 and count up to the 10, how many steps did I take? Or, I can use Cuisenaire rods: what is the difference between the orange rod and the light green rod?

For place value, I can use counters again, possibly using the “exploding dots” version, or an abacus. Or I can use base 10 blocks, which help the students see successive powers of ten geometrically. Or I can have students make their own base 10 materials using beans and tongue depressors.

For multiplication, I can use counters again: 3 times 5 can be represented as 3 sets of 5 counters. Or, I can arrange the counters in a rectangular array, which allows me to see that 3 times 5 equals 5 times 3, and previews the area model for multiplication. Or I do 3 hops of length 5 on a number line. Or I can learn to count by 5’s— a condensed version of the number line model.

The power of math is precisely that the same structure (in this case whole number arithmetic) describes many different phenomena. Understanding these representations, and their relationships with each other is a lot more powerful than teaching this subject in a single way. It respects students’ intelligence, and it is more true to the math: numbers are indeed all those things, whether you like it or not.

Is this confusing? If it is, it is because numbers are challenging, not because we approach them in multiple ways. If a teacher is comfortable with all the models, uses them strategically, and helps students think about the connections, multiple representations are illuminating, because they reveal the many meanings of the underlying math.

The same is true about fractions: the widely used “pie” representation provides a great connection to angles and to time as seen on a clock. A number line representation emphasizes that a fraction is a number, and reveals its relative size. A rectangle representation facilitates computation and helps to introduce the idea of common denominator (as you will see below).

What prompted me to write this post is that I recently expanded a page on my website about a particular representation of fractions that I suspect is not sufficiently used in upper elementary school: rectangles on grid paper. One question I was asked was whether it would be confusing to students to see yet another representation of fractions. I say: quite the opposite! This is an accessible model that can be the foundation of important discussions, and can be used for many different fractions topic. (It’s a fractions knife!) Of course, it should complement (not replace) other approaches. Here is an example from that article: which is greater, 2/3 or 3/5?

Using 3 by 5 rectangles (as suggested by the denominators) makes it easy to see both fractions as parts of the same unit. Counting reveals that 2/3 is 10 squares, or 10/15, while 3/5 is 9 squares, or 9/15. Which is greater has been made clear. Those very same rectangles can help us think about 2/3 + 3/5 or 2/3 – 3/5. Read the whole article (and watch my homemade videos) here.

Or take the use of letters in algebra. Sometimes, *x* is an unknown, and to find its value we learn to manipulate symbols. Sometimes we use it to make a general statement about algebraic expressions, for example 2*x = **x *+ *x. *Algebra manipulatives can help with that. Sometimes it’s a variable, and we see what happens to an expression as *x* varies. Tables, graphs, and function diagrams can help us understand that. Sometimes *a, **b, *and *c* are parameters. They constrain an expression, but they are not unknowns or variables.

A well-intentioned but pedagogically clueless mathematician once commented that variables are easy to teach: just patiently explain to students that they behave just like numbers! In fact, the reality is that it is a good idea to use many representations: tables of values, graphs, manipulatives, symbol manipulation, and function diagrams. Done well, this is not confusing: it’s illuminating. The historically catastrophic failure rate in the traditional Algebra 1 course is largely caused by the naive belief that patient explanations of how to move x’s and y’s on a piece of paper is the only way to go, as long as you follow it by mind-numbing and meaning-free practice.

Or take trigonometry: the sine can be seen in a right triangle, or in a general triangle, or in the unit circle, or in a Cartesian graph. This is not because a teacher decided that. It’s in the math itself — and it gives us multiple approaches to the concept, with opportunities to make connections between them. (Here’s one such connection.)

**Conclusion: ***Multiple representations reveal the many-fold meanings of math concepts*. To make that less confusing, the answer is not to limit students to a single representation, even if you’re proud of how good you are at that particular one. Instead, give some thought to a proper sequencing of the representations, encourage students to use the representations that makes sense to them, and especially be sure to connect the representations to each other.

Anyway, one of those books was *Polyomino Lessons *(free download.) Polyominoes are the figures one can make by joining unit squares edge-to-edge. The five tetrominoes consist of four squares each, and are the raw ingredients for the game of Tetris:

There are twelve pentominoes (five squares each), which are used in many, many puzzles of all types, some of which are included in my book *Working with Pentominoes,* which is available from Didax.

One question I asked my students (and myself) was: what is the minimum figure that can cover all the polyominoes of a given area? For example, this figure works as a minimum tetromino cover:

The figure has area 6, and can accommodate any of the tetrominoes. No figure of area 5 would work.

What I love about this problem is that it is fun and interesting to students as well as teachers. Or, at any rate, I find it fun and interesting.

Recently, I read an excellent article by Patrick Honner about minimal covers. It is not related to polyominoes, but it did remind me of the minimal polyomino cover question. A little internet research yielded nothing, so I decided to see if I could figure something out beyond what I found in the 1970’s and 80’s. Back then, I concluded that the minimal covers from monominoes to hexominoes had areas 1, 2, 4, 6, 9, and 12, respectively. See if you can find those covers. If you tire of the problem, or to confirm your solutions, look at pages 17 and 47 of *Polyomino Lessons*. I also included the same exploration in Lab 4.7 of *Geometry Labs. *(Answers on page 189.)

To go beyond that, I decided to find a minimum cover for heptominoes (7-ominoes.) This turned out to be more challenging than I thought. After several false starts, I ended up understanding the question a little better, and found a cover with area 17:

I am confident that no square can be removed from this shape: every single one is needed by one or another heptomino. I don’t think a shape with area 16 would work, but I don’t have a proof that such a shape does not exist. So at this point, I think the sequence is 1, 2, 4, 6, 9, 12, 17.

If you find a heptomino cover with area less than 17, please let me know! Likewise, if you can prove me right that 17 is optimal, and/or if you explore this question with a computer program.

Thanks in advance!

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Minutes after I posted the above, Benjamin Dickman replied on Twitter:

17 appears to be correct: oeis.org/A327094

Specifically, this was solved on code golf: codegolf.stackexchange.com/q/167484

In response to an earlier MSE question: math.stackexchange.com/q/2831675

Thanks, Benjamin! I had actually looked for the sequence on OEIS, but didn’t find it because when I looked I thought it was 16 for heptominoes! I should have looked again once I convinced myself about 17… Or I should have done it the way Benjamin did it. He searched for:

“1, 2, 4, 6, 9, 12” polyomino

Still it was a fun exploration, and I got to share the question with math teachers! Certainly the problem up to pentominoes or hexominoes is suitable for a math club, math circle, or math team.

]]>Figuring out an approach to any substantial topic takes time and effort, and that is rarely shared publicly. Nolan Fossum’s presentation is the sort of session I’d love to see more of at conferences. He shared his current approach to teaching conic sections — as a work in progress. His goal is to have less memorization, less reliance on formulas, and more intellectual engagement from his students. As a participant, I really appreciated being invited into his process. (He frequently acknowledged his approach needs work, and he encouraged us to send him our suggestions.)

I will not try to summarize the whole talk, instead, I’ll focus on one wonderful thing I learned. Let me set the stage for it.

In the construction unit I created for my geometry class many years ago, I challenged students to construct points equidistant from a point F and a line *d*. One such point (V) is easy to find after dropping a perpendicular from F to *d*:

Soon after that, students realize that drawing a parallel to *d* through F helps to find points A and B, equidistant from F and *d*:

This is correct, as A, F, and D are three vertices of a square, and likewise B, F, and D.

At that point, I tell students that there are many more points equidistant from F and *d. *If necessary, I suggest that choosing a point T on *d* may be helpful. Where are points that are equidistant from F and T? How does that help us? This leads to the construction of a generic point P, which is equidistant from F and *d*:

The locus of P as T moves on *d* is the parabola with focus F and directrix *d*. (That, of course, is the geometric definition of the parabola. Read more about this here, and here.)

(Here is the GeoGebra file I used to make the above figures.)

Back to Nolan’s talk. He calls points A and B the “anchor points” for the parabola, and he asks: what is the slope of VB? I had never thought about this. If you are familiar with the construction of V and B (as in the second figure above), it is easy to figure out, and it is the same for all parabolas! Once you know this and the coordinates of the vertex, you have a way to find the coordinates of one of the anchor points. This in turn means that if you know the coordinates of the vertex, you have a straightforward way to locate the focus and the directrix without a lot of memorizing. I love this!

There are other geometric constructions of the parabola, of course, but this one is particularly important because it throws light (heh) on the parabola’s reflection property. (Again, see this page on my website.)

This very same construction came up later in the day. Along with many others, I tried to attend another talk on conic sections. Here is an excerpt of the session’s description:

…construct conic sections as the curve of symmetry between a circle and a point inside the circle, an infinitely large circle and a point, and a circle and a point outside the circle, revealing how the ellipse, parabola and hyperbola

relate to each other.

The presenter never showed up, but a few of us stuck around, and tried to make sense of this. I had never heard the phrase “the curve of symmetry”, but I like it. I realized that the “infinitely large circle” mentioned in the summary is line *d* in the above construction. That led me to see that by following the same steps I outlined above, but putting F inside (or outside) a circle *d*, we should be able to create an ellipse (or hyperbola.) It was very satisfying to use GeoGebra to see that indeed this worked, and even more so to prove this was correct. Here is the figure I made:

See if you can prove that the curve is indeed an ellipse. (Where are the foci?) Dragging the red point outside, it turns into a hyperbola. The proof is similar.

(Here is the GeoGebra file.)

I co-presented “Lessons from Lew” with Kim Seashore. (Read more about Lew Douglas here. Get all the handouts, plus other relevant materials here.) As is usually the case when presenting something for the second time, it went really well. I was really pleased at the participants’ creativity in the kinesthetic part of the session: they came up with excellent representations of 60° angles, complementary angles, etc… Later, when discussing the Symmetric Polygons handout, Dan Bennett astutely noted that we used “at least two lines of symmetry” for the equilateral triangle, but did not try for a formulation with fewer than four lines of symmetry for the square. (This could have been “at least three lines of symmetry”, or “at least one line of symmetry through vertices, and one line of symmetry not through vertices”.) Logically, he has a point, but he agreed that pedagogically this was probably the right call.

Later, I presented “Connect the Dots”, which was a combination of curricular lessons and assorted puzzles using the geoboard (or dot paper.) I have presented the curricular part many, many times, and it always goes well. More recently, I ran some Math Teachers’ Circles focusing on teacher-level challenges. Combining all that into one talk turned out to be a great idea. Kate Philpott pointed out that the puzzles I intended for teachers could work as extensions for students, even in fifth grade. Of course she’s right. Including those puzzles helped to make the session relevant to ages 9 to 99 — as advertised.

So yes, I had a great day. Alas, I didn’t get to take my usual walk along the coast, so I’m already thinking about what I should present next year. Whatever it is, I’ll be sure to take that walk!

— Henri

]]>On the other hand, merely providing good explanations does not turn out to be any more effective, as students don’t necessarily listen to those, and even if they do, they may not understand them. Thus, in order to remember things, they are forced to memorize poorly understood techniques and ideas. Because there is so much math to be learned, memorization often only works until the quiz. A few weeks later, it’s mostly gone.

As a teacher, I had to learn how to combine student inquiry with teacher guidance. I reject the hardcore “never give a hint” position of people who overestimate inquiry. I also reject the belief that excellent teacher explanations suffice. Effective teaching requires us to navigate between many strategies which appear to be mutually exclusive, but are in fact complementary. Well-chosen problems help students engage with the question at hand, and prime them to listen to and understand teacher explanations. How to choreograph this back-and-forth dance is learned through practice, and through teacher observation and collaboration.

Unfortunately, a culture of “I explain it, then you practice in silence” still dominates too many math classes, leaving no room for student intellectual engagement. In this post, I will share pedagogical bits and pieces which I used as part of a *guided inquiry* approach. I do not claim these add up to a full scheme. Rather, they are components that can be added to any teacher’s repertoire. I hope they can help you move from the lecture + practice paradigm towards active student-centered learning. And if your class is already discovery-based, you may find here some techniques to add to your toolbox.

Students cannot always hear the answer to a question they don’t have. A consequence is that even after a great explanation from the teacher, they may not absorb what they have been told. If instead of starting with an explanation, you start with a question to get students to think about the topic, they are much more likely to understand your explanation.

For example, ask students to find the area of graph paper triangles (or parallelograms, trapezoids, etc.) before mentioning that there is a formula for that.

There are many benefits to this. It may reveal strategies that make more sense to the students than the ones you come up with. It gives students the confidence they can recreate the formula if they don’t remember it. It makes “base” and “height” concrete and visible rather than numbers on a figure. A discussion of student solutions can lead to a discussion of how to generalize what was learned, leading up to a formula. And of course, starting this way does not prevent you from providing your own explanation later.

Many traditional math class questions are, well, boring. What is 7 + 3? What is the greatest common factor of 12 and 15? Graph y = 6x + 4. Solve 6x + 4 = 3x + 10 for x. And so on. Reversing the traditional question often yields great engagement and powerful discussions. Here are some examples.

**Find pairs of numbers that add up to 10**. Even answering this with positive whole numbers gives more students a chance to contribute answers. But this can expand in many directions, depending on the class. You can ask for *sets* of numbers that add up to 10, such as 3, 3, and 4. If you are getting started with integer arithmetic, great patterns will emerge if you allow negative numbers in the pairs, and your students will be practicing their new skills as part of an interesting quest. Likewise, in Algebra 2, if you ask that same question about complex numbers.

**Find pairs of numbers whose greatest common factor is 3. **This is sure to reveal what understanding your students have about common factors. If some students don’t understand the underlying concepts well enough to do this, they can get help from their classmates, or from you. If even that is not sufficient, some review is in order!

**Find the equations that will yield these graphs. **

This is both more fun, and more educational than “graph this, graph that, what do you notice?” (More about this.)

**Find equations whose solution is x = 2**. Instead of the standard “simplify” process, students get to “complicate” the equation. This gives them an opportunity to practice “doing the same thing to both sides” as part of a creative challenge.

All these examples show the power of reversing traditional questions. On the one hand, it makes it possible for stronger students to show off their fancy solutions, and to remain interested and engaged. At the same time, all students can find their own level of challenge, and you can support them as they take that next step. This is a sort of differentiation with no artificial ceiling: everyone gains.

When I first started teaching in a high school, I was given a terrible piece of advice: “Aim for the middle”. Our program was not tracked, and we had a wide range of talents and backgrounds in each class. Since that is difficult to manage, the idea of aiming for the middle seemed like common sense: if you aim too low, you are betraying your stronger students; if you aim to high, too many kids will be frustrated. As is often the case, common sense was not a good guide.

Since then, I have developed some strategies to handle a wide range of students in the same class, which I share in Reaching the Full Range. One key component of these strategies is to *aim high*. In everyday class work and homework, by all means include some material that is too easy, and some material that is too difficult. But for the activities and challenges that anchor an important concept, it is best to be ambitious. If it turns out the question is too difficult, it is always possible to support students with hints, or some other kind of scaffolding. If it is too easy, it is not moving the class forward, and it is giving the wrong message. Here are some examples.

**Rich anchor problems. **Start a new unit with a Big Question that encompasses the concepts you are about to teach. For example, to prepare students for sequences, explore what happens when you iterate the function *y* = *mx* + *b*. In other words, if you start with a certain value for *x*, and it yields a *y*, use that *y* as the next *x. *Repeat and see what happens. This is especially interesting when *m *is between -1 and 1, but all cases are worth exploring. A full discussion provides an opportunity to introduce subscripts, to think about limits, and eventually to zero in on the usual arithmetic and geometric sequences.

**Add another representation**. Do not limit yourself to a single approach to your topic. Look for other ways to think about it. Can manipulatives provide a productive environment to explore these ideas? Can technology help? Is there a visual representation that could be helpful? For example, use well-chosen graph paper rectangles to illustrate the addition of fractions:

In this example, a 3 by 5 rectangle stands for the number 1. That choice makes it easy to represent the fractions we are adding. By examining the figure, we see that the sum is 13/15. Doing this once, of course, would not be very helpful, but discussing the dimensions of useful rectangles, leads to a powerful strategy which can be applied to comparing, adding, or subtracting fractions.

**Have students make things.** In some cases, a *constructionist* approach can yield much learning. Constructionism is a theory of learning espoused by the Logo movement in the 1980’s, and it is still going strong with Logo descendants such as Scratch and Snap, or with the current popularity of maker spaces and STEM. To enhance math education, what students make needs to have a curricular payoff. It can be a turtle geometry computer program, a GeoGebra construction, a tiling of the plane, etc.

For an example of the latter, using a plastic template and a pencil (or a computer), students can create tessellations using triangle or quadrilateral tiles. Here is one based on a scalene triangle:

This figure offers a context to discover and discuss many basic geometry theorems: sum of the angles in a triangle, exterior angle, parallels and transversals, translations and rotations, …

Pursuing any of these suggestions should enhance discourse in a math class: who talks? who listens? who does the thinking? The key is to keep the students at the center. We, their teachers, can help, but they are the ones who need to do the learning.

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