"Mathematics talk" for five year olds

Question

I am trying to prepare a "mathematics talk" for five year olds from my daughter's elementary school. I have given many mathematics talks in my life but this one feels very tough to prepare. Could the members of the community share their experience with these kind of lectures. I was thinking to talk about some theorems of Euclidean geometry which will include some old fashion compass, straight edge construction with some kind "magical outcome" and then try to give kids some logical reasons for the "magic". Any ideas?

Edit: I would like to thank one more time to the members of the mathoverflow community for their generous input and support as well to report the outcome of my talk.

I just got out from my daughter's elementary school where I ended up teaching four class periods today instead of the one I originally prepared for. I taught two sections of 5 year olds (26 kids a section) as well as two large group of fifth graders (close to 100 kids in total). I was "over prepared" to talk to 5 year olds which came handy with fifth graders.

Inspired by the answers from this forum I chose to talk about Platonic solids and have kids mostly engaged in practical activities as oppose of "teaching" them. My assistant chair at Augusta State University Georgia has generously shared her large collection of POLYDRON blocks. I had three bags full of equilateral triangles, squares, and pentagons. I have also pre-build one set of all five Platonic solids (Tetrahedron, Cube, Octahedron, Dodecahedron, and Icosahedron). I have also printed out cut and fold maps for all solids from this website and gave them to kids together with building blocks.

We identified first the properties of polygons (number of sides, vertices, and angles) of each of building blocks were to use as well as the fact they were regular (sides and angles of equal length). I was rather surprised that five year old children have no problem identifying pentagon as it is the shape of rather important building in Washington DC.

Then we introduced the rules of our "game":

1. Only the same "shapes" were to be used for building solids.

2. Two faces could meet only in one edge.

3. Each vertex of the solid had to meet the same number of faces.

Five year old kids had no problem assembling Tetrahedron, Cube, Octahedron however not a single group (they were allowed to work alone of in groups of 2-3) was able to assemble Dodecahedron, and Icosahedron. This was not the case with fifth graders (older kids) where several groups (4-5 out of 100 kids) successfully assembled Dodecahedron, and Icosahedron.

Even 5 year olds were able to identify number of faces, edges, and vertices by counting from the cut and fold charts. They had harder time identifying Schläfli symbols for each Platonic surface due to the fact that they had to count them on my pre-build models but they have never the less accomplished the task. We were able to come up with Euler characteristic (magic number as I referred) but the real focus was on subtracting numbers which we did using our fingers. Obviously the kids got lost after the cube due to the size of numbers involved. I was not able to convey any information about further combinatorial properties of Platonic solids related to Schläfli symbols to five year olds.

On another hand fifth graders had no problems identifying

$$pF=2E=qV$$

but had hard time solving equations as $pF=2E$ for $F$ and $2E=qV$ for $V$ and substituting into

$$V-E+F=2$$

not a single fifth grader could follow my computation for the estimate

$$\frac{1}{p}+\frac{1}{q}>\frac{1}{2}$$

where were effectively ended our little lecture.

In both sections kids asked me to preform some more "magic tricks". I glued a long strip of paper for them creating a cylinder and Mobius bend. Many kids thought of cylinder as a circle and Mobius bend as a figure eight (few fifth graders mentioned infinity symbols) even that they could not give any logical explanation why they think that way. We cut cylinder and Mobius bend and children start cheering my name when Mobius bend "broke" into just another bigger Mobius bend.

Five year olds wanted to hug me after the lecture and sit at my table in cafeteria. The fifth graders were either indifferent or came to me after the talk to shake my hand and ask if I can teach another class. It is also worth noting that while playing with blocks many fifth graders made prisms, pyramids while some try to pass non-platonic solids for Platonic solids bending rules of our game.

Teachers were trusty for this kind of experience. They are in a bad need for professional development after years of budget cuts and fear for their jobs. The school is going to buy blocks. I hope to make visits semi-regular and help them as much as I can (obviously out of selfish interest to improve my daughter's education). I have already planed to introduce some other games like tangram, pentominoes, and Hanoi tower. I will also install GeoGebra on their computers.

I might edit this post in next few days and add few details.

Most Kind Regards,

Predrag

Answer 1

I'm going to quote Bill Thurston from his interview for More Mathematical People:

Thurston: ... One thing that is very important is the education of children... In the elementary schools in Princeton that my kids have attended, there is an annual event called Science Day. They bring in scientists from the community, and we spend a day going around from class to class talking about things. I have enjoyed doing that quite a bit.

MMP: What have you talked about?

Thurston: I have done different things every year for ten years or so; for example, topology, symmetry, binary counting on fingers... I find that kids are really ready to pick up mathematics in the way that I myself think about it. Of course, it's toned down.

MMP: Can you be a little bit more specific about the way you think about mathematics?

Thurston: That's a tough question. It might be nice to give an example. At one time I went into a class of kids and made lots of equilateral triangles. We made a tetrahedron by putting three triangles at each vertex. Then I asked what happens if you put four triangles, and they constructed an octahedron. Then with five triangles at each vertex they constructed an icosahedron. But with six triangles they found that the construction just lays flat. And then I asked about seven triangles at each vertex. They pieced it together and they got these hyperbolic tesselations in four-space. They loved that. The kids did. But the teacher really felt ill at ease. She didn't know what was happening.

Answer 2

I've seen a very successful interactive math talk to a participating audience of five and six year olds at MSRI a few years ago. It was structured around the question of "what's the largest number?" Kids had fun coming up with large numbers but eventually one of the youngsters figured out a simple way to always come up with a number bigger than the previous one named. Through some nice leading by the teacher, they eventually concluded (indeed proved) that there are an infinite number of natural numbers. It was fun watching the kids (by then fully engaged) grapple with what is really a quite abstract idea --- that one can know that the numbers are unending without actually exhibiting them in any way. It was a terrifically sneaky way to engage kids and hear their ideas on notions like "infinity" or even "number".

Answer 3

Giving a little "Magic show" about Möbius strips might be fun. Make a huge number of Möbius strips and cylinders, and hand them out to the kids along with safety scissors. You could have a predrawn "center line". Ask them what will happen when they cut along the center line: How many pieces will you get? They will probably say two for both shapes. Have them cut along the line and see what happens! The result for the cylinder is as expected, but for the Möbius strip you get a piece of paper with two twists. Now ask them what happens if they cut this in half. You get two interconnected links! You could have them start with a new Möbius strip and cut halfway between the center line and one edge, following their way around. You get a Möbius strip linked to a double twisted strip! This will all be great fun for the kids.

You can "explain" some of these phenomena by showing them how to make a Möbius strip themselves: just take a strip of paper, twist it, and tape the ends together. From this perspective cutting a Möbius strip in half is just the same as taking two strips next to each other, twisting both of them, but the head of one piece attaches to the tail of the other, so you can see how cutting the strip in half only leads to "one piece". It might help to have some different colors of paper, so they can more easily keep track of the "two halves".

You could show them some Escher drawings involving Möbius strips - the one with the ants would probably delight them.

Answer 4

I think you'll have the best luck if you try to make it interactive. Kids that age have very low attention span and very high energy -- they like to use all their senses, so I would avoid just talking at them for any period of more than five minutes. I would also avoid general logical reasoning, which in my experience can't really be grasped at that age. What's possible, though, is to go through enough examples (or have them go through enough examples!) to give them a feel for why something is true, without providing any kind of strict argument.

But what really makes an impression (like Henry says above) are patterns, especially ones that have pictures associated with them. I've had good luck drawing out Sierpinski's triangle (and having them draw it too, which is fun), then introducing Pascal's triangle, then coloring the evens one color and the odds another and seeing Sierpinski's triangle pop out. If you can get them to realize that one can just do Pascal's triangle all the way working mod 2, then it will be an amazing success -- and if they can actually understand why Sierpinski's triangle shows up, it will be a miracle.

Answer 5

I addressed my son's class in school when he was five. it was not quite the same thing as what you're asking about: it was part of a series of "what I do at work" talks by parents, and it was very brief. I wanted to give a little taste of topology. Of course I prepared a big Moebius strip and did tricks with it. I considered also counting $v-e+f=2$ for some convex polytopes, but I decided to keep it simple, so as not to overreach or overstay my welcome. So instead I drew an octagon on the chalkboard and had them discover that this thing with eight sides also has eight corners. And when that had sunk in I remarked that if I had drawn something with one hundred sides, it would have had one hundred corners. One excitable little boy shouted "Do it! Do it!" So, one conclusion: yes, they do like big numbers.

Answer 6

I have put my book, Modern Math for Elementary Schoolers, on the Internet. The book is based on my teaching experience at LAMC, Los Angeles Math Circle, a free Sunday school for mathematically inclined children, currently second grade through high school, run by the UCLA Math Dept. I had taught my own son and a few of his buddies using the material that later became that book from the first day of their kindergarten. You can find the book at the following URL.

http://www.naturalmath.com/DeltaStreamMedia/OlegGleizerModernMathematics_12_2011.pdf

The book is copy-lefted. You can use it for any non-commercial purpose.

Answer 7

Here's a short list of activities that could be fun to try:

Begin by giving the numbers 1 through 9 to 9 students. Here it's good to have a physical number to give them -- a piece of paper with the number written large will work. Ask them to line up in order. Ask 7 whether he or she is even or odd (you don't have to remember their names if they are holding up numbers). Ask 7 about the numbers next to her -- are they even or odd? (5-year olds won't automatically know that even numbers are surrounded by odd numbers. They might not know the meaning of even and odd until you run the activity.)

Further activity: Have only 1-5 stand up in order. Then rearrange them using only transpositions (say "Number 2, switch with Number 5"). Have the students count each transposition. Then have a (well-chosen) student try to put them back in order using only "switches". How many switches did it take? Can 5-year olds discover the sign of a permutation? How about if you record the number of switches and point out even/oddness?

Further activity: Have only 1-5 stand up in order. Have them shake hands in pairs and try to have the others count the handshakes. How many handshakes were there? An even number or an odd number of handshakes? Can 5-year olds discover how the parity of "n choose 2" depends on n?

Further activity: Have the numbers 1-9 stand up in order again. Have them find a partner to add to 10. Then back in line again. Have the even numbers step forward, and the odd numbers step back. Then the even numbers back and the odd numbers forward. Then back in partners to add to 10. Are evens partnered with evens? Odds partnered with odds? You can ask lots of questions and keep the kids moving.

Watch out -- you might have to bring extra numbers and modify groups so that all kids can participate.

Good luck! When in doubt you can ask the 5-year olds why 6 is scared of 7.

Answer 8

On three occasions I surprised 4 and 5 years olds with counting on one hand to 10. here's how this is done: http://www.mathteacherctk.com/blog/2010/07/counting-on-one-hand-and-on-two/

The kids knew to count fingers in the conventional way. They could not believe it is possible to go beyond that. They tried and, when this worked, they were delighted. For a talk, I would first show that there are several ways of counting to five: bending/straightening fingers, starting with a thumb or the pinky. I would stress the point that however you count the result is always the same. After that I would count to 10.

You can prepare chocolate bars and then ask how many breaks it would take to break them into squares. It's a different way of counting the squares so the result is also the same however you break the bars: http://www.cut-the-knot.org/proofs/chocolad.shtml

Another fine activity has to do with the braid theory: draw vertical lines, join them randomly with several horizontal lines, and then follow from top to bottom alternating vertical and horizontal lines, changing direction at every horizontal endpoint: http://www.cut-the-knot.org/Curriculum/Algebra/Shuttles.shtml Perfect for job distribution among the kids.

Answer 9

If there are more than 26 kids in the group, you can tell them you know for sure that two of them have names starting with the same letter (more than 12: birthday in the same month, and so on). In a column in the newspaper, the physicist Robbert Dijkgraaf recounted how he did this once with a group of five or six year olds; one of them got it immediately, and a lively discussion followed.

From there, you might try to explain the pigeonhole principle, or go on to counting, or even do the thing about people knowing each other at a party (though that might be too hard).

Good luck!

Answer 10

I ran a bunch of classes at a school a while ago. My students were older, but lessons to be learnt:

• Do craft: I did lots of origami. This is beyond 5 year olds. But what about this: attach a pencil to a string and then attach the string to a point on a piece of card. Now observe that by drawing then pencil with the string taut you get a circle. If you fix the string in more places you can get different shapes. You'd need to do a fair bit of prep for this of course.... You could also do something connected to symmetry. Use a mirror and ask what it means for two things to be the same. Distorting mirrors could be used for comparison. Drawing half a butterfly in wet paint and then folding it in two to get the other half. That sort of thing.

• Do magic: I did a binary numbers one: I got someone to think of a number and then showed them cards and asked them if the number was on the card or not. Then after 8 of these cards I told them the number. The trick was that each card corresponded to the numbers 1 mod 2, 2 mod 4, 4 mod 8 etc. Even just finding a number on some card may be too hard for five year olds but the principle of presenting your material with some theatre is sound: it can make anything appear to be a little magical.

• Come in character: You could be the Numbers Wizard or King Triangle or something. Wear a cape covered in numbers, have a pet rabbit called Cubey, make moo-ing sounds. Whatever feels right to you :-)

And the suggestions above for moving around a lot, avoiding `giving a talk' as such but being interactive are all totally sound.

Good luck!

Answer 11

I've spoken about the "puzzles" that Terry Tao and I developed for Schubert calculus, like the left two here:

I handed out pieces (the 0-triangles, 1-triangles, and rhombi) for the 3rd graders to assemble, in groups, telling them to make triangles. Then made a table with n = #edges on a side (any side, since they're equilateral), k = #1s on a side (theorem: any side), n-k, #1-triangles, #0-triangles, #rhombi.

Different groups made different puzzles, and I included some little ones (n=0 and 1) in the table. Then asked if anyone saw patterns. I got the answers I wanted, which were that #1-triangles = $k^2$, #0-triangles = $(n-k)^2$, #rhombi = $k(n-k)$.

It works nicely with younger children, too, but they're less likely to guess these formulae.

Answer 12

See "Picture-Hanging Puzzles" by Erik Demaine et al. (arXiv link):
          

Answer 13

Many five-year-olds, given a pair of compasses, will use it to stab their neighbour. But the better-behaved ones should be able to manage this:

Draw a circle.
Centre a point on its circumference, same radius, draw an arc within the circle, running from circumference to circumference.
Centre a point where this arc meets the circle, same radius, draw another such arc.
Repeat.
By magic, after six arcs, you end up where you started.

Answer 14

A very similar question was posted just over a year ago on Reddit's /r/math by a professor who had to speak to his daughter's class about what a math professor does. He was speaking to first graders so some would have been six, not five, but that's close enough.

Later, he posted a followup reporting how it went.

Summary: fractals, especially the Mandelbrot set. The kids went absolutely wild over it. They grasped the self-similarity. Parents told him their kids came home and would not stop talking about the Mandelbrot set.

Answer 15

Computer Science Unplugged offers plenty of possibilities:

CS Unplugged is a collection of free learning activities that teach Computer Science through engaging games and puzzles that use cards, string, crayons and lots of running around.

The activities introduce students to underlying concepts such as binary numbers, algorithms and data compression, separated from the distractions and technical details we usually see with computers.

Of course, some might argue that Computer Science is not a subset of Mathematics...

Answer 16

I did the following with 7 year olds several times when my children were in elementary school, and it might work with 5 year olds, too, if they know how to add. (Although maybe enough to know how to count.) The topic is Triangle Numbers and Square Numbers. First we played with triangle numbers $3,6,10,15,\ldots$. I drew them with dots on the blackboard, and the children, split into groups of 3 or 4, modeled them using M&Ms. Then we discussed how to get the next triangle number from the previous one, leading to the formula $T_n=1+2+3+\cdots$. (Of course, I didn't write this as a formula, but they seemed to have no trouble grasping the idea of putting another layer on the bottom of the triangle.) Next we turned to square numbers $4,9,16,25,\ldots$. Again, with pictures and M&Ms, they easily understood what a square number is. Then came the challenge. How to efficiently compute $S_n$, keeping in mind that although the children knew how to add, they did not know how to multiply. The solution, of course, is that $S_n=1+3+5+\cdots+(2n-1)$ is the sum of the first $n$ odd numbers. This becomes clear from the picture if you label the dots in shells. Here's a $5\times5$ picture using letters, but in the class I used colored dots, and the children made their own M&M models of a $4\times4$ square with the colors to illustrate the shells: $$\begin{matrix} E&E&E&E&E\\ D&D&D&D&E\\ C&C&C&D&E\\ B&B&C&D&E\\ A&B&C&D&E\\ \end{matrix}\qquad 25=1+3+5+7+9$$

After all this fun, I posed the real question: Are there any triangle numbers that are also square numbers? So we made a short list of triangle numbers and a short list of square numbers and found that $36=T_8=S_6$. After this triumph, each group took 36 M&Ms and used them to transform $T_8$ into $S_6$, and then they got to eat the M&Ms.

To wrap things up, we tried to find another square-triangle number. Each group was tasked with making a list of either $S_n$ or $T_n$ by repeated addition, then we compared the lists. My recollection is that this was not always succuessful due to arithmetic errors, but that was okay. (The next one is $1225=T_{49}=S_{35}$, then $41616=S_{204}=T_{288}$.)

I've also talked about this subject to high school students (without the M&Ms), leading to Pell's equation and more-or-less proving that there are infinitely many square-triangle numbers. And also to college students, proving that the square-triangular numbers form a "1-parameter exponential family", i.e., that Pell's equation has a unique generator. This is one of the reasons that I like this problem so much, it can be studied at so many different levels.

Update 2024: When I recently did this activity with a grandson's second grade class, I wasn't allowed to bring in M&Ms. Not sure if it was the possible issue with food coloring (which I'm now more aware of, since I have a valued colleague who is quite allergic) or whether feeding children candy in school is frowned upon (also a reasonable concern). In any case, we used pennies instead of M&Ms, which worked fine.

Answer 17

I would definitely not agree on giving a "talk" to 5 yo kids. It is very difficult to keep their attention for more than a couple of seconds. Unless, you tell a story.

The topic I would choose is "counting", and being more specific "counting by ordering"

Example 1: Make a slide with 15 dots in random positions, ask them: How many points are there?

Then a second slide with 15 dots, three groups each of 5 dots arranged as in the face of a dice. Make the same question as before.

Then a third slide with 15 dots arranged in a rectangle (if they can multiply this is easy 3x5). again make the question.

The moral is that by ordering counting is easy.

Example 2: Make them walk in the room, and ask them: How many kids are in the room? But don´t let them stop walking... -Do you want to stop?- then make them stop and ask them again... this should be much easier (even for an adult)

You can include some problems on counting (like those where you have to count how many triangles are there in a given picture)... you know, there you need to be careful not to count more than once... or make them draw a couple of lines in a piece of paper, then mark the intersection points and when they're finished ask them: -How many points are there? -how many triangles?

... well I guess you got the idea.

wish you luck!

Answer 18

Keep it fun and interactive. Some Game Theory could go well. Rock-Paper-Scissors will rock, if you can program some toons to play different strategies...

Maybe some problem/puzzle solving, say some River crossing puzzles or Rubick cubes (if you get one rubick cube for each kid), or mathstick puzzles, if Rubick cubes are out of your budget :-))

Answer 19

When I was an undergrad, I heard a story where a young child was excited by watching 6 equal-sized equilateral triangles fit together to form a regular hexagon. I don't remember what her age was, but this sounds doable for 5-year-olds, especially if you make the triangles take the colors of the rainbow, excluding indigo.
This is exceptional, in that this is the only example of a regular polygon decomposable as the finite disjoint (except for boundaries) union of smaller regular polygons of a different shape. If one drops the "different shape" requirement, one can put equilateral triangles together to make a bigger equilateral triangle or put squares together to make a bigger square.
But 5 may be too young to get a feel for how, for example, angles work. I don't know how they'll handle failing to put equilateral triangles to make a square, for example.

Answer 20

I was quite successful with the basic variant of game Nim among individual children.

Answer 21

perhaps they could play with a bunch of these

http://www.youtube.com/watch?feature=player_embedded&v=VIVIegSt81k#!

Answer 22

I gave a lecture to the Wrexham Science Festival some years back on "How mathematics gets into knots", advertised as for 8-80, but I think it extends. You see some ideas for this on the knot exhibition part of this site.

Things you can do are:

Dirac string trick (using the home made apparatus apparatus illustrated there, two wooden squares, one with an arrow on it, coloured ribbon, and bulldog clips to fasten the ribbon to the board, easy to undo in case everything gets tangled), and related to the belt trick and the Phillipine wine glass trick (do a search on this, and also on Air on the Dirac String). We have found young children love this, but best to let them try an empty glass or plastic mug first!

Showing addition of knots is commutative, using just a piece of rope. Hope that helps.

Update: A flat model of the Mobius Band is easy to make and fun. Do a Google search on "Brehm Model". Here is a link to a transformation of this into a sculpture.

Another thing for kids is to cut out and make Borromean Squares. Again, do a web search on this. Even Borromean triangles.

Answer 23

There's a specific math trick I was very fond about, that I had concocted myself as a child (I guess I was 10 or so, but I'm sure most 5 year olds could grasp it). I remember impressing/puzzling not just peers but many adults. However, it's not possible to show the child and still have a puzzle for yourself. In a sense it's the opposite of what you seek, a trick simple enough for a child to perform, that can still puzzle most adults the child would meet.

Basically I claimed I could read people's mind. I would allow the other to use a calculator or a piece of paper out of my view. It takes the form of a dialogue:

Child: "I can read your mind, just select a secret random number between 1 and 100"

Adult: "Oh really? OK, I have number in my mind" (suppose 17)

Child: "add 7"

Adult: "OK, I added 7" (24)

Child: "Multiply with 9"

Adult: "Sure, let me think" (216)

Child: "Now add all the digits together, if the result has more than one digit, add them together again"

Adult: "Done" (9)

Child: "Add 4, and concentrate on the result"

Adult: "OK! I'm focussed" (13)

Child: (spooky nonsense, feigned concentration, and other charlatanery) "13!"

The main lesson is that when you multiply a number with 9, all the digits add to a multiple of 9, so repeatedly adding these ends up with 9 itself. When doing this with different people you can hide this, or distract with variations. So practically anything before multiplying with 9 is irrelevant, and after adding to 9, you just calculate along. Instead of multiplying with 9 you can also multiply with 3 twice, or multiply with 8, then add the previous number etc to hide the dependency on this step when doing the trick multiple times.

Considering how much I loved to amaze people with this cheap dirty trick (sure, they didn't all know much math, but most people can use a calculator), I predict most children would enjoy doing this to others too. I suggest you first try the trick with the child, and then once the child is convinced you're telepathic, you break the spell and explain how it works. But then you'll have to suffer your child fooling all the family and visitors for the next couple of years..

• The sequence of operations before the x9 step, is practically irrelevant, but best kept simple, especially when the other is not using a calculator, because they might be using a large number.
• The x9 step can be obfuscated with variations. Now both parties are synchronized on 9.
• The sequence after the x9 step is irrelevant, but you have to follow each operation along. This last phase is better when lengthy, so people forget about the x9 step. For example starting from 9: minus 2 (7), square (49), times 2 (98), minus 25 (73), "73!"

Answer 24

Ask 10 children to create the Desargues's configuration. Each kid should be a point from the picture. It is easy only on a paper!

Answer 25

I had a positive experience teaching a few groups of elementary school kids (of mixed ages and abilities) in Uganda a lesson plan centered around Euler's formula, focusing on graphs (there are several great answers here suggesting lessons for surfaces). I think most of it will be suitable for younger kids too.

The formula is simply:

$$\chi(\mbox{Graph}) = \#\mbox{Vertices} - \#\mbox{Edges}\;\; (= \mbox{"one minus the number of loops."})$$

I preferred dimension 1 over 2 because my goal was to lead the kids through the process of mathematical discovery, taking as little as possible from their sense of agency and independent exploration.

The simplicity of dimension 1 means:

1. The kids can easily generate many examples (on the board or on paper).

2. There are only two, more accessible (easier to count), ingredients in the formula. This makes the discovery of the relationship possible with only gentle steering (e.g. I suggested we tabulate the counts for some examples; I provided the first couple and they were trees, so the kids typically continued to generate $\chi = -1$ graphs for the initial set of examples).

3. Once the case of trees is established, you can explore graphs with loops. This is both rewarding and important (a hint of homotopy invariants, see also the next step). With surfaces, this would involve going to higher genus (or non-orientable surfaces...). This is certainly possible (and perhaps worthwhile if you decide to do surfaces) but for me it seemed too technically challenging to explore without applying too much "force" in directing the kids' play.

4. You can see what happens to the invariant under simple transformations of the graph (contracting an edge or adding a vertex in the middle of the edge). I was hoping the kids would discover a proof of invariance along these lines, but we usually ran out of time before we got there (I think it's still worth a shot though).

P.S.: Two more recollections: a. At least some groups explored disconnected graphs: $$\chi(G) = \mbox{"number of connected components"} - \mbox{"number of loops"}.$$

b. We counted the number of "regions" you could color-in. I think this was mostly a way to define what we mean by "the number of loops" more precisely in step (3), but this can also lead naturally to the formula for surfaces: if you include the region at infinity, the blackboard essentially becomes a sphere (of course this only works for planar graphs, but all the graphs kids draw are planar. I suppose one could try to discover the more general definition of "number of loops", that would make the formula work for non-planar graphs! This might require some straws or strings, to build non-planar graphs with, or you could start from a cell structure drawn on a torus and examine the 1-skeleton graph)