math recommendations for a 3 year old

I was recently asked for suggestions by a parent of a 3 year old.

There are a lot of different resources I could suggest, but they really depend on the child and the parents. The main question for customization is about the parents: what are their starting assumptions about math/math learning and how much do they want to engage on selecting/planning activities?

For example, if a parent doesn't really get the growth mindset, I would advise a heavy dose of Jo Boaler. If the parent wants open explorations and can build their own specific tasks, maybe the Vi Hart videos are good inspiration.

That aside, there are a few resources/products good enough that I’m willing to give blanket recommendations:

1. Lots of tools for measuring. Playing with measuring has so many benefits, I can’t list them all, but some of the highlights are (a) seeing math and numbers all around us, (b) tactile engagement, (c) inherent process of comparison, and (d) natural connection with language as the kids and parents talk about what they are measuring/why. The links I've provided just show examples, I am not necessarily recommending them over other versions.
1. Set of plastic measuring cups (imperial units and fractions)
2. Tape measure (we just used standard adult tape measures, but as a recommendation, you need to be careful about tape measures that have fast return springs for cutting or catching small fingers)
3. Balance scale and set of standard weights (this math balance is a good option and one we bought)
4. Timer (we liked this one)
5. For older kids, a step counter, GPS wrist-watch showing speed, thermometer, pH meter, electricity meter are all interesting additional measuring devices.
2. Talking Math with your Kids:
1. E-book
2. Blog. I recommend reading all the posts, I think they are a superset of the material in the e-book, so this is a better resource unless you want the “curated” highlights. This link goes directly to posts tagged 3 years old.
3. Tiling toys and shapes book in the TMWYK store. I particularly like Which on doesn’t belong? A better shapes book.
3. Denise Gaskin’s Playful Math books: these talk about general habits and methods in an intro section, then specific activities (mostly games) in the rest of the book.
4. I got a lot out of these storybooks (free to print) with my kids: CSMP Math Storybooks.
5. Standard gambling tools: playing cards and dice (I like pound-o-dice for the assorted colors, sizes, shapes)

There are some computer games/systems, a lot of board games, and mechanical puzzles, but the stuff above is where I think parents should start for young children.

What do you think of my recommendations? Any additions you think are worth adding to make a top 10?

Chocolate cake reference

This was Nigella Lawson's recipe, but that page currently doesn't include the ingredient amounts, so I'm reproducing it here. This was a test ahead of J3's birthday, so she helped. Since it seems successful, we will be making it again and I will include more pictures of the process.

Heat oven to 170 deg C

MIX:

• 3/4 cup + 1 tbsp flour
• 1/4 tsp salt
• 1/2 tsp baking soda

1/2 cup water (boiling)

6 tbsp cocoa powder

COMBINE into a thick paste and allow to cool (doesn't need to come fully to room temp)

2 tsp vanilla extract

ADD to cocoa paste and stir

2/3 cup olive oil

1 cup sugar

3 eggs

COMBINE in mixing bowl and mix on high (directions said 5 minutes, we mixed for 2-3)

COMBINE wet ingredients and mix thoroughly

COMBINE dry mix and wet just to incorporate together.

Use small amount of oil to grease a 9 inch baking pan.

Pour in batter and bake for 45 minutes (I rotated the cake every 10 minutes).

Another dessert reference to store, unfortunately dairy: Caramel Sauce

Ode to a bead string (a non-poem poem)

from The Math Maniac

What is a bead string?
100 beads
Whites and Reds
5+5+5+5+5+
5+5+5+5+5+
5+5+5+5+5+
5+5+5+5+5
From hand and eyes to brain
But also: a caterpillar, a snake,
dog collar and horse bridle
a gun

a gate or a fence

a telephone line
a part of a train
lock for your door
tiny eggs (maybe spiders?)
coins
jewelry (necklace, naturally, or earrings, bracelet, belt)
roller tracks for a made-up truck
beans cooking in a pan
and so much more!

---

While reviewing Dreambox Learning, I spent some time thinking about physical manipulatives compared to apps. Apps can certainly have nice features, but physical objects own my heart. I'm not even going to talk about the educational value (see this Hand2Mind note for a summary and further references). The thing we really like is the open-ended flexibility about how they can be used, both mathematically and for creative play.

As an example, the bead string is (maybe?) one of the more limited maipulatives because it is linear and one connected piece. We don't even have one. However, the ideas above were immediate ideas coming from observations of how my kids play and a quick brainstorming session with the little ones.
BTW, we find that a 6 to 9 year old can make any object into a gun, in case that idea didn't make sense to you.

Favorite category of math manipulatives: food! In this case, freshly baked chocolate cake.

Returning to math, one of the cool things about this open play with math manipulatives is that it provides a lot of easy entry points into short math chats. These are fun, in themselves, and also reinforce the observation that math is accessible and all around us.

Manipulatives in Apps
While better than nothing, virtual versions of physical manipulatives always seem to fall short, with the following limitations:

• only one "correct" way to interact with them
• no possibility of combining
• can't take them apart

Perhaps there are also some benefits of virtual objects. In particular, they don't need to obey the law of conservation of mass. Can that, or other advantages, be used cleverly to make up for the disadvantages?

Let me know what you think and tell me where you've seen the best use of virtual manipulatives.

Start of vacation math

Here in Thailand, the kids are now on summer vacation, so I've been exceptionally busy and found it hard to write blog posts. We've still been doing a lot of math, so I wanted to at least note down some of the activities.

Sonobe Modular Origami

Credit for this one goes to another parent at J1's school. One day when school was still in session for J1, but J2 and J3 were already on vacation, we happened to see a group making sonobe modules at the elementary school. This is something my mother tried to introduce to the kids last year, but it didn't catch them at the time. Not sure what was different this time, but J2 was really intrigued.

Most of the work has been J2, but J3 also got into the act and folded about 9 units herself. I helped to sharpen the folds for 6 of them, which J2 and J3, working together, made into a cube.

Here are two of J2's stellated octahedra, made in between swimming sessions at the beach:

Video references we found useful for guiding our sonobe exploration:

Sonobe Cube: this is where we started.

Stellated Octahedron: shows how to make the basic module and how to assemble 12 into a stellated octahedron. This proved to be easier than I had anticipated, so J2 was able to do this all on his own.

Doubled tetrahedron: unfortunately, this doesn't give a sonobe module tetrahedron, but rather a 6 sided shape. We had actually made one of these ourselves when experimenting with 3 sonobe modules created from post-it notes. These are delightfully strong and feel really nice to toss from hand to hand.

Some advanced observations
Euler Characteristics
The model for the cube gives a triangulation for which it is particularly easy to count vertices, edges, and triangles. I talked about the Euler characteristic and we then did the counts for other sonobe models as well as other platonic solids to check. We also talked about the Euler characteristic for a cylinder (without end caps) and a torus.

Finally, I liked to describe our calculation as "0 dimensional things" - "1 dimensional things" + "2 dimensional things." This led J2 to ask what we would get if we included the space inside the shapes (extending to a "3 dimensional thing") and to get curious about 4 dimensional objects (which led to a lot of discussion, below).

Chirality
Sonobe modules come in two flavours, depending on which corners you fold into the center. All three of us (J2, me, grandpa) accidentally made the two kinds of sonobe modules without realizing it, only to struggle when we went to assemble a larger model. This started a brief conversation about chirality. While we didn't explore in depth, it did spark a later moment of recognition when Matt Parker mentioned the concept for mobius strips (see below).

Robustness
I was really surprised that the sonobe models we built are so robust. The completed shapes don't fall apart very easily (especially the doubled tetrahedron). Even better, for the models we have built so far, the sonobe modules themselves don't need to be perfect. This was crucial for the beginning origamists in our gang as it allows them to do almost all the work from start to finish.

It also made me think about my old idea of error in construction recipes (straight-edge and compass constructions as well as origami folding constructions). To illustrate this for my father (and J2 unintentionally) I showed them David Eisenbud/Numberphile's construction of the 17-gon. This, in turn, prompted J2 to get excited about straight-edge and compass constructions (below).

Constructions

As mentioned above, J2 was excited about investigating classical construction problems. This is something he has seen a couple of times in the Abacus Math curriculum that P does with the kids. However, in that, we haven't generally been strict about the tools they use and the collection includes some standard triangles (45-45-90, 30-60-90) and measuring tools (rulers, protractors). Now, J2 was curious to see what could be done with just a straight-edge and compass.

For physical construction, our tool of choice has been the Safe-T compass:

Obviously, this doesn't quite correspond to the ancient limitations, but we ignore the ruled markings and the various holes are always within our margin of error on radius lengths anyway. Other than the obvious (for a parent) advantage, I think this compass style makes it easier for 2 people to help each other when drawing the circular arcs.

J2 has also been playing Euclid the Game. The virtual constructions have added two interesting points:

• strict restriction to what is constructed; he eyeballed a pair of equal lengths at one point, then wanted to talk about why he hadn't gotten credit for the construction (for angle bisection). This actually helped him identify that he needed the lengths to be equal; once he articulated that, he knew to use the compass to draw a circle.
• Chunking into advanced tools. For example, the game gives you a mid-point tool once you demonstrate the construction of a midpoint of a given line segment. This has helped him start to work on the coding concepts of functions/procedures and also gives him a new visual reference for the problem solving strategy "try to use what you already know."

Fourth dimension

The big insight here is about the relationship between understanding four dimensional shapes and our own ability to perceive 2 and 3 dimensional shapes. Other than that tip, I'd direct you to these great resources:

Flatland: we don't actually have a copy, but I know it well and it is a foundation behind a lot of the discussions we had and videos we watched.

Matt Parker's Royal Institution talk: He covers a lot that isn't about four dimensional shapes, but I'm not going to point you to the relevant starting time because the whole talk is great!

XKCDhatGuy: This was actually one of the first videos J2 found and, hey, the kid does a pretty good job explaining. Don't, by any means, stop with this explanation, but it is a fine place to start.

Carl Sagan's Flatland: J2 liked it, but I still remember the old cartoon from the BU library that we would watch at PROMYS and wasn't so impressed.

4th dimension explained: [no comment]

Drawing 4th, 5th.. dimension:  Somehow, the simple points in this video really helped demystify higher dimensional shapes for J2. I think part of the point is that it encourages us to attend to aspects of the shapes that we can understand easily (for example, the number of vertices in an n-dimensional cube) rather than worrying about the macro shape.

Rotations of 4 dimensional shapes:  Similar message to the previous video

There is no 4th dimension: Nice, quick explanation living up to the high standards of other One Minute Physics videos. A very worthwhile series, in its own right.

Cricket

This isn't mathematical, but I wanted to record this for posterity as well. Through random chance, J2 asked, "How do you play cricket?" At this point, he is still 2/3 English, having spent most of his life in London, so I guess he has a right to some information about this crazy pastime. For better or worse, though, that's not what he got. Our house rules:

1. batsman/batsmen pretend to stand in a traditional cricket batting pose, mime holding a bat
2. I stand a couple feet away, mime delivery of a cricket pitch (straight arm overhead throw)
3. we all (including spectators) run together into a giant heap, tickling each other and saying vaguely English-y, vaugely cricket-y things: "good show old chap," "ooh, looks like rain," "how do  you take your tea?" "you were almost to a century, young chap" etc
4. I break off the tickling and everyone stands in line, waiting for their points and an explanation of why points were awarded. Not spilling the tea is the most common reason for someone getting points, though boundary hits are also frequent. Note that points can be awarded in non-integer amounts (1/3, sqrt(2), and pi have all been awarded at some point).
5. Finally, I declare the match over with a result of "no result." As you can tell, we take our cricket seriously and are committed to the "test match" format.

Some important summary points:

• J2 is the player who has managed to accumulate the largest score in a single match. He was awarded "one hundred million one thousand" points by J3 for not spilling the tea.
• We are coining a new English-y phrase: "young old chaps." Usage example: when J3 is setting up a chasing game outside, she will point to J1 and J2 saying "these young old chaps are the runners," then point to me "and that old young chap is the monster."

Assorted math conversations

A collection of little items from the last month.

Optimal time to eat

P was talking with the kids about what order they should eat their food, which led to lots of graphs showing flavor over time. I don't know if this is a typical conversation for other people, but pretty common in our house. Here are some of the charts they made: Naturally, this also led to other charts. In particular, a debate about how quickly someone could develop skills by practicing music. Note the lack of labeling on the y-axis.

Some challenges

At lunch today, everyone got a math question. Apologies, I don't remember what challenge we gave to J1. J3 was given what is 5 + 2? After we let her consider and answer, J2 got asked what is the cube root of 125? Before either older kid could answer, J3 yelled out: 5.
Hmm. No, we also don't think she really understood, but it is nice to dream!

A joke

J2: look, a prime! (pointing generally at the 100 chart on our wall)
Everyone look expectantly at the wall
J2: No, it isn't. 22 is a compositive
All little Js burst into hysterical laughter. P looks at me as if to say, "this is your fault."

I wonder

From nowhere: Daddy, what 3 digit number has the most factors?
Exploration to follow tomorrow.

Cooking

J3 and I made scrambled eggs today in two batches. First, with milk for J1, using 2 eggs. Then, a second batch without milk for J3 using another 2 eggs.

J0: how many eggs did we use
J3: For J1, we used 2 (holds up two fingers on right hand). Then, we used two for me (holds up two on left hand). Let's count them . . . 1, 2, 3, 4. Four eggs!

Contortionist cubes

We got a lot out of this video from Mathologer: Contortionist cubes.
An immediate idea was to see which of the objects we could construct using our materials at home. The best we managed was using polydron to build one of the contortionist cubes, with sides that are either all red or all green. This high-tech enterprise was made possible by scotch tape!

Little one counting

Late last year, Joe Schwartz wrote a very interesting post about the difficulties one student is having with counting and skip counting. I recall hearing a theory that many later math difficulties trace back to when a student missed solidifying the concept of one-to-one correspondence and some related concepts of counting (but don't have a citation or reference). As a result, I was thinking about how our little J's understand these ideas.

In particular, I wanted to try out Marilyn Burn's little game from one of her comments with J3:

Ask him to put out 8 cubes on a paper. [I chose 8 because when I remove one, the child won’t be able to know how many by subitizing.]
Ask: How many cubes did you put on the paper? (8) [Here I look for whether the child has to recount.]
Say: Watch as I take away one cube. Remove one cube and place it on the table.
Ask: How many cubes are there on the paper now? (7) [Does the child have to recount, or does the child just know.]
Say: Watch as I take away another cube. Remove one of the 7 cubes and place it on the table.
Ask: How many cubes are there on the paper now? (6) [This is the same as the previous question, a way to check if the child still needs to recount.]
Say: Watch as I put one cube back on the paper.
Ask: How many cubes are on the paper now? (7) [Similar, but adding 1.]
Sometimes I repeat again removing a cube and asking: Can you tell me how many there are without counting? Some kids shake their heads to indicate they can’t, others say they’ll give a guess, some are able to.

J2

Before I had a chance to try out the question sequence with J3, I had some time alone with J2. He was sorting colored pencils, so we used those as counters. Overall, he breezed through the questions, but there were two amusing points:

1. After separating out 8 colored pencils, I asked how many he had. His response showed that (a) he believes in conservation, so he knew there would be 8, but also (b) he is used to me doing something tricky, so he wanted to verify that there were still 8.
2. His method of verification: split them into two groups of 4, an amount he could recognize by subitizing, not counting.

He asked me why I was asking these questions and I told him it was related to his understanding of hierarchical inclusion. We talked briefly about what that means and he was delighted by the term, so ran off to ask J1, "how is your understanding of hierarchical inclusion?"

J3

My counting time with J3 came during dinner. She was eating cucumber slices, so we used these as counters. This turned out to be a mistake, since conservation doesn't work with edible counters! In other words, whenever I asked her how many slices were in our cluster, she would pop one in her mouth and smile, knowing that she was teasing me.

Mainly, though, I was able to verify that she doesn't yet have the concepts that allow short-cuts to the questions in Marilyn's sequence and needed to recount to get all the answers.

Incidentally, I started the activity by telling her that we were going to count something together. She immediately grabbed this coaster and then accurately counted the circles to 37.

For me, the entire experience was a really interesting illustration that counting actually requires a complex collection of sub-skills.

Surreal numbers and whole body integers

Two unrelated activities to note:
(1) Exploring checker stacks and surreal numbers with J1 and J2
(2) A whole body numbers game with J3

Surreal Numbers and Kids

If, like me from one month ago, you don't know about surreal numbers, I think you'll find they can be a really engaging exploration with kids. The main attraction is the appearance of infinities and infinitesimals, both of which really seem to resonate with young mathematicians. In addition, there's fantastic icing on this cake, too: you can explore by playing a simple (to learn) game with a lot of depth.

Credits: this exploration is strongly inspired by Mike Lawler's recent posts about surreal numbers and the Jim Propp post that inspired him. If you are interested in doing this type of exploration with your kids, I strongly suggest going through all of their posts on the subject.

Note: since we used black and red checkers, while the convention in the other posts is blue and red, I will abbreviate B and R so you can naturally substitute your own preferred color scheme.

How we got started
Using a set of regular, stackable, checkers (black and red), I showed each of the older J's the position RB + BR (a stack with red on the bottom, black on the top and a stack with black on the bottom, red on top) and explained that the basic moves.

This was a good initial example because it let us talk about each of the major scenarios:

• We will investigate cases where B moves first and others where R moves first
• Each one can only take stacks above one of their own color checkers
• If the colors allow, they can take a top checker and leave the rest of a stack undisturbed
• If their color is the bottom of a stack, they can remove the whole stack
• Usually, they will have choices about which stacks to remove

Then, I explained the losing condition: if you don't have any more moves, you lose. They quickly realized this was the same as when they no longer had any checkers of their own color on the board.

Next, we quickly played a set of simple games:

1. Single B checker
2. Single R checker
3. B + R
4. RB + BR
5. RB + B
6. B+B+B+B +R+R (and similar)
7. BBB+R+R (and similar)

Connecting with numbers

I told them that one amazing thing is that we can give each game position a value. Then went through:

1. Single B checker is +1
2. What do they guess a single R checker is? explain -1
3. What about B+R? eventually get to 0
4. Explain the fundamental trichotomy: positive value means winning strategy for B exists, negative for R, what about 0?

Powers of 1/2

The first really juicy bit came when I asked what they thought the value of a BR stack would be. This established a common sequence of investigations:

1. If alone, can we see a winning strategy for either B or R? In this case, obviously B. Thus, the value is positive
2. Compare with 1 by playing the game BR + R. They were really fast about seeing this link, for others this is worth writing out and spending some time discussing. In this case, they saw BR + R must be negative, so the value of BR is between 0 and 1.
3. Guess a value; they've got enough experience to think that often the answers are "nice," so 1/2 was a natural guess.
4. Test. In this case, it meant checking BR + BR + R

Next stop was RB. I wanted to make sure negatives weren't left out and to reinforce the symmetry in the colors, so would ask them to swap the colors and get a value as a quick follow-up.

Next, I asked them to see if they could find a configuration with value 1/4. This took a long time and there were lots of false starts. I didn't think this would be easy and didn't help them shortcut the exploration.

Once they got 1/4, though, they had a fast guess about 1/8. We checked it and then they made a conjecture about further powers of 1/2.

Deep blue and red

Seeing powers of 2 in some form is always fun, but both knew that they had been promised infinity and wanted to see it. Using our plastic dinosaurs, I introduced the deep checkers deep blue (represented by a blue mini) and deep red (represented by an orange mini).

Our sequence for these was similar to the powers of 1/2. The value of deep blue is positive, so they compared to 1 by playing the game BBBB.....+R. Each had a sparkle of insight, but quickly played BBB.....+R+R+R just to check, then announced that the value would be bigger than any integer.

I gave them the name omega, and we checked the deep red: RRRR.....

At this point, one of them put the deep red on a B checker and asked what that would be. Again, followed the previous recipe to realize that it must be positive, but smaller than 1, then smaller than 1/2. At that point, one of them made this arrangement:

Ok, I know that mastadons aren't dinosaurs

If it isn't clear, the realization was that the game sequence B, BR, BRR, BRRR, ... would have values

1, 1/2, 1/4, 1/8, .... and BRRRR..... would be at the end of that sequence with value ... larger than 0, but smaller than every power of 1/2. You can see that they made a similar connection with the negative values.

A little bit about deep purple

The two kids were totally satisfied now, having gotten omega and epsilon (along with omega +1, epsilon - 1, omega + epsilon, etc). To give them something to chew on for later (and because we had purple dinosaurs in our set) I introduced the deep purple, BRBRBRBR.... Immediately, they designated the yellow dinosaurs as the inverse of deep purple (RBRBRBRB.....)

So, what's the value of deep purple? What they've gotten so far:

• positive
• smaller than 1 (by playing BRBRBRBR..... + R)
• Bigger than 1/2 (by playing BRBRBRBR..... + RB)
• guess 2/3. I don't know where this came from, but I confirmed that is the value
• working on finding winning strategies for the second player in R+R+3(BRBRBR.....)

Wrap up videos

As a round-up in the evening, we watched the videos from the first post in Mike Lawler's sequence. I paused frequently to let my two shout out their answers and explanations for where Mike's boys were in their exploration. This seemed to be a very effective way to underline their experience for the day.

Number match on the number stairs

A game for J3. This is a simple game with some variations that makes use of our stairway "number line" and a three year old's natural enthusiasm for running up and jumping down stairs.

As pictured previously, we have labeled the stairs in our house from 0 to 36 (more to come). We have a set of cards with numbers on them. P mixes them up, then gives them, one at a time, to J3 to put on the corresponding step, and they sing count up and down:

Some other variations:

• using playing cards instead of number cards
• Using cards with dot or shape patterns
• Using cards with number words ("one" instead of "1")
• child sends the parent to a particular step, checks if the parent got it right

Pillow forts

In case you missed it, pillow forts have been in fashion recently. Here is an example:

war variations

Most of you have probably seen how the standard card game "War" can be modified to make an arithmetic drill game. Denise Gaskins probably has the best description here: Game worth 1000 worksheets.

We have used three variations of this game a couple of times: straight War (J3 and J2 playing with greater than, equal to, less than), addition war (grade 1), and multiplication war (grades 2 and 3). Frankly, I am often surprised how enthusiastic the kids are to play, since there aren't any choices for them to make when they play. For those who are ready to move on from the basic game mechanic, here are some extensions and related explorations.

Extension games

Build your deck
Currently, Vanguard, a deck building game, is very popular amongst the Js. One possiblility for War is to let the players arrange their deck in advance. In a sense, this is like a more granular version of rock-paper-scissors. I particularly like this variation for the 2 (or more) card versions where the kids need to think about how to mix high and low value cards. Also, the number of cards burned on each War battle can upset the organization for the rest of the deck, so that adds a layer of complexity for them to consider.

Choose your cards
My favorite variation is to deal a hand (between 3 and 6 cards, replenished after each "trick") to each player and then let them choose which ones to play. You can either require simultaneous play or, as we prefer, have each person play one card at a time going around clockwise, like in Bridge.

Explorations

Some exploration questions:

1. In basic War (high card wins): will there always be a tie at some point during the first pass through the deck?
2. In basic War: can there be a complete game (one player loses all their cards) without a tie ever occuring?
3. Does basic War always end with one player losing all their cards or can there be cycles?
4. How many times do we expect a tie on the first pass through the deck?

All of these questions can be explored for the different variations. For elementary kids, these are very challenging questions and I don't expect many answers. Two recommended ways to explore:

• Play many games, record data and observations. Make conjectures and see if there are any counterexamples that disprove your ideas.
• Play a simpler version of the game by reducing the number of cards in the deck. For example, play a demonstration game with only 6 cards: A, 2, 3 for two suits.

Modern math competition

Saw this brief exchange and thought it would be a good time to write down some ideas for a new type of math event:

There is a STEM Expo later this year w/ community partners and a 4th - 8th gr competition. Ideas on how to make comp more mathy? #MTBoS

— Jana Sanchez (@jsanchezmath) December 18, 2015

It will be clear below that these are all other people's ideas, I've really just added ideas for how you might score this.

Fold and Cut

Everyone gets a pair of scissors and three pages with a straight-line figure (a shape that has a boundary made up of polygons) in the middle of the page. For each shape, players try to find the smallest number of straight cuts they need to cut out each shape.

One example of a shape to cut out

Another example: cut the house and the tree at once (hard!)

Scoring for each shape:
10 points for least cuts
5 points for second
1 point for all who cut out the shape successfully, regardless of how many cuts it took

Reference: Fold and Cut Theorem.

Estimation

Set up an estimation challenge (number of marbles in a jar, weight of a collection of books, height of a blown-up picture of your favorite mathematicians, etc). All players write down:

1. Their estimate of the amount
2. An explanation of their reasoning
3. A value that they think is too low
4. A value that they think is too high

Scoring:

Each estimators performance is based on the absolute size of their error (|estimate - actual|) multiplied by the size of their low-to-high range, divided by 2 if the actual value is within their range. The lower the resulting value, the better.

20 points for best performance

10 points for second

5 points for third

1 point for everyone who estimates

Reference: Estimation180 has tons of great prompts for estimation.

Which one Doesn't Belong

Really simple: show players four things, then they figure out reasons why each of the figures could be the one that doesn't belong.

This is just a small sample of the ideas for Which One Doesn't Belong

Scoring:
1 point for every different reason + 5 bonus points if there is a reason given for each of the 4 shapes

Reference: Which One Doesn't Belong has a huge collection of great ideas for shapes, numbers, graphs.

Notice and Wonder

Pick a 3 to 5 math pictures, gifs, or videos to show on a large screen. Players write down the things that they notice ("I notice that ...") and wonder ("I wonder [what, why, when, how many, ...]..?").

From SolveMyMaths

Scoring (similar to WODB):
1 point for every notice or wonder + 3 bonus points for having both a notice and a wonder.

Reference: SolveMyMaths and MathHombre both have really nice collections. Mathematical Etudes has some excellent videos that don't seem to be as widely known among Anglophones. I'm sure there are others

Big numbers and BIG NUMBERS (emulating classic TMWYK)

During a recent chat with J3, 3.5 years old, I decided to test some ideas about her number sense. She was playing with 3 and had just counted them.

me: How many fingers do you have?
J3: 10, here's five (shows one hand) and here's five (shows the other hand)
me: Let's count them?
J3: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
me: I'll try mine. 1, 2, 3, 4, 5, 7, 8, 9, 10, 11
J3: (laughs) That's silly. You skipped ... you don't have 11. Let me show you. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
me: Oh, thanks for helping! I have something else to show you. (I write down the numbers below). Do you know which is bigger?

Why is it so dark? Oh, yeah: winter

J3: this one (points to 306).
me: what number is that?
J3: 3-zero-(pauses) 9.
me: what about this one?
J3: 3-(pauses) 8

I  realized that I had drawn the 306 a bit larger than the 36. Maybe that was why she pointed to 306? I drew two more.

me: Hmm, what about these two numbers?
J3: (points to the 2). That is a really big number (stretches her arms wide)

we then drew faces for each other.

A J2 project and challenge

J2 was arranging the pieces from a checkers set. He wanted me to pose this challenge to all of you:
what are your favorite ways to arrange 24 checkers, split equally into two colors? here is one of his designs, to get you started:

We are working on imitating JJ Abrams lens flares, but only have gotten lens glare

counting at the pool

who: J3
where: at the pool

---

In the spirit of Christopher Danielson's marshmallow post, Tale of Two Conversations, I wanted to flag some simple ways to add some numbers to a toddler's day. None of these are earth-shattering and they aren't hard to do, but they sometimes take a bit of awareness to remember.

Counting up

At the pool recently with J3, I noticed several times that we chose to use numbers when we might have said something else. First, she started jumping in from the side, so we led with a count: "1, 2, 3 jump!"

That's standard and comes easily to most parents (my observation from watching other parents at playgrounds). A math educator once told me that some kids start school thinking that the counting sequence is "1, 2, 3, go" because they hear that formula so often!

One of our additions is to add a little counting song after each jump. I always think of the sesame street count, so we sing after each jump: "1 [later 2, 3, 4, etc] mighty jump, ha ha ha." She got up to 17 before moving on from the jumping game.

Counting down

Later, she did rocket launches from the ladder. Of course, this was a good chance for counting down practice: 5, 4, 3, 2, 1, 0 blast off!

Oh, look at the time!

Wrapping up all of this, we had a series of comments about the time throughout our swimming session. When we first arrived, I read the time to her, with a couple explanatory comments (it was 10:55, so I explained that the long hand on the 11 meant 5*11 = 55 minutes after the hour and the short hand near that 11 meant that it wasn't quite 11 o'clock, so it would be 10). I didn't expect or require her to get all of this detail.

Then, we talked about how long we could stay: 15 or 30 minutes? 30 minutes was the choice, then when would we leave? As the time passed, how much time did we have left? When we got to the agreed leaving time, how much more time did she want?

I found that the conversation about time, in particular, had to be a conscious effort. Since she doesn't have a strong understanding or awareness of time, it would have been just as functional for me to keep all that information to myself rather than explaining it to her. On the other hand, how can the little ones develop their own understanding if they aren't part of these types of discussions?

more age chat

who: J1
when: bedtime

---

J1 wanted to return to our discussion about their three ages. In particular, someone had noticed that, prior to the recent birthday, their whole year ages were all primes: 3, 5, and 7. J1's question this evening: will this ever happen again?

First case

We talked through 2 different cases. The first uses 2 year gaps, so AgeY(J3)+2 = AgeY(J2) and AgeY(J2)+ 2 = AgeY(J1). As we worked through examples, J1 quickly saw that it wouldn't work whenever J3's age was even, so we focused on odds. From 5, 7, 9, he realized that we could skip the cases where J3 was 7 or 9, so we looked at 11 and 17. Along the way, we had identified that J1's age was 9 (3x3), then 15 (3x5), then 21 (3x7).  Seemed to be a pattern of always being a multiple of 3...

Second case

Next, we looked at their current gaps, so AgeY(J3)+2 = AgeY(J2) and AgeY(J2)+ 3 = AgeY(J1)
This one was easier, solved by just looking at the parity of the ages.

So, sad news all around: they will never again have all prime ages!

Fractional ages

Next, we talked about the equation Age(J3) + Age(J2) = Age(J1). This works out now when we use whole year ages (or using the floor function, if you prefer). There had been some chatter earlier about fractional ages, so J1 wanted to check that.  These were the estimates J2 had given earlier for their ages:

• J3: 3 1/4
• J2: 5 1/2
• J1: 8

So, does the equation hold? Unfortunately, it doesn't. J3+J2 is 8 3/4. Will it ever hold in the future and when?

We talked for a while and J1 realized that, as time passes, the sum of the sibling's ages increases twice as fast as his age does. That gave him a clue that it won't happen in the future, but did in the past.  So, when was that?

Keeping in mind the pattern he'd recognized about changes, he guessed it was 3/4 of a year ago. Checking through was a good exercise in fractions and confirmed his conjecture.

From that, he wanted to get more precise about when that date was, 3/4 of a year ago, and then started talking about that having been a really special time for the three of them (unrecognized at the time, of course). When he started getting that precise, though, I offered that J2's estimates were a bit off and substituted J3 = 3 1/6, J2 = 5 5/6, and J1 = 8. This gave an even more satisfying conclusion once he worked out this case.

Freeform tangrams: an imagination game by J1

Who: J1, J2, and J3 (a late appearance)
When: just before bedtime
What did we use: two sets of tangram pieces

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J1 came up with a new game this evening. The rules:

• number of players: 1 or more
• playing pieces: 1 or more sets of tangram pieces
• designer: each round, one player takes the role of the designer, putting the tangram pieces into a configuration. This could be a random arrangement or intentional
• Taking a turn: in a clockwise order starting with the person to the left of the designer, players say what they think the tangram looks like from their perspective. After everyone has taken a turn, the players shift 90 degrees to their left and repeat the cycle, but this time they are looking at the tangram from a different orientation.
• Once everyone has had a go from all 4 orientations, the next player becomes the designer
• Advanced play: once someone has said what they think the tangram looks like from their perspective, the experienced players can add a comment to start building a story based on that object.
• Winner: everyone!

We played with J1, J2, and J0 taking turns as the designer. To give you a feel for it, here is one of our designs from 3 different orientations:

Here are some of the ideas for what we saw: person jumping out of a box, a wheelbarrow, an army officer, a flower, an 8th note in a mirror, a catapult, a road grader, a knife cutting an apple, the sword in the stone, I was surprised that no one said "poop," which just shows how engaging the two of them found this activity. I doubt it would have been as successful if I introduced these rules and asked them to play!

Our own puzzle book

Some comparisons (two tmwyk transcripts and a puzzle)

who: J1
when: just before bedtime

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the value of being alive

J1: Daddy, now I've got a question for you
J0: Ok?
J1: if I get a new book every time I write 20 pages in my journal, how valuable is each page?
J1: The books are about 150 baht
J0: How much?
J1: let me check, I think the price is on the back cover . . . 169 baht
J0: if you could tell me what I need to calculate, I'll calculate for you
J1: hmm, so 20 pages is 169 baht, I want to know how much one page is, so I need to divide by 20.
J0: do you think it will be more or less than 10?
J1: less than 10
J0; Are you sure? How do you know?
J1: Well, 10 * 20 is 200 which is more than 169
J0: what about 5 baht per page? Is it more or less than that?
J1: More, 5* 20 is half of 10*20, so 100, which is less than 169.
....
<we figure out that the amount per page is 8.45 baht/page>
....
J1: That's not very much!
J0: How much did you get for your birthday?
J1: [x] from grandma, [x] from grandpa
J0: well, how much is that per day. Is it more or less than 10?
J1: More than 10
J0: how <interrupted>
J1: how do I know? well . . .10 * 365 is ...
<some discussion of whether he was right, various other estimates of the amount of money per day>
J0: How does that compare with each page of your journal?
J1: More...but what if I include the [present a] and [present b]?
...
<he estimates how much different presents cost, figures the total, estimates how much that is per day, etc>
...
J3 (who has been listening all this time): wow, J1, that's a lot of money!

J3 explores bricks

Earlier in the evening, J3 has been building sticks with 1x1x1 TRIO cubes. She made four, all the same length, then handed two to me as drumsticks. I counted the cubes in one (I got 11) and then she counted one of hers (she got 12). I put them side-by-side and we saw they were the same length.

J3: but...daddy, I really counted 12, you are wrong
J0: are you sure they should have the same number.
J3: yes, let's count them again, together
<I point at the cubes and she counts them, 11>
J3: Ok, now I'm going to build a shape and you see if you can make a copy. It will be tricky!

A birthday puzzle

With their current ages expressed as whole years (you know, the way everyone talks about ages, except for mothers of very small children):

1. What is a number sentence that relates the ages of J1, J2 and J3? Hint, oldest is 8, middle 5, and youngest 3
2. Will this ever be true again?
3. Was it ever true in the past?
4. When/why not?
5. What about multiplying? Will it ever be the case that AgeY(J1) = AgeY(J2) * AgeY(J3)?
6. Was this ever true in the past?
7. When/why not?

Note that there is a complication since they were not all born on the same day, so the difference in their year ages changes depending on the day of the year we are considering.

J2 wanted to investigate more precisely, so he asked to work things out in months. That meant we had to calculate how many months are between them.

We are totally flipping out!

Who: J1, J2, J3
when: dinner time
What did we use: 4 coins

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In Sue VanHattum's linkfest post, there was an intriguing puzzle: given four coins in a square, can you call out commands to flip coins so that you are guaranteed to eventually get them all up or down? More precise instructions/rules are in the link.

We saw very quickly that this is an excellent puzzle:

• complex terminology? Not at all.
• difficult to find equipment? Nope.
• low threshold? Yes, Even J3 was able to investigate along with the older ones
• high ceiling? Yes. Solving the initial puzzle is hard enough, but there are plenty of possible extensions. 
• Fun and deceptively tricky challenge? YES!

Everyone got to have fun taking a turn as the coin master (following the flipping instructions and rotating the square) and most of us took a turn as the (attempted) solver. Here are our tools:

For most of the conversation, I just listened. The kids naturally focused onidentifying states: what are the possible conditions for the coins, which of these are, for purposes of the puzzle, identical and which are truly different? One other part of the discussion was about moves, again, which are identical and which are different. Of course, the didn't quite use this terminology.

Some extensions

Simplify?
I suggested they try the puzzle with a smaller number of coins. 1 coin, no problem, it always starts solved. 2 coins were pretty easy again. What about 3 coins? Here, there was a little thinking about what the equivalent version would be for 3 coins. In particular, do the coins have this configuration:

and the coin master can only swap the outer two coins, or are they in an equilateral triangle like this,

and rotations are the only transformation?

Well this is math, so there are no right or wrong answers, you just have to try out your different ideas and see what is the most beautiful.

Another simplification we thought about was to eliminate the rotations.

Probabilistic Attack vs Strategic Game
The problem asks for certainty, but let's assume the rotations are done at random. What if  you are happy with a high probability of getting a solved configuration? Is there a different strategy that has a faster expected ending time? Can you say anything about confidence levels (probability of ending at n or fewer moves)?

In contrast, what if the coin master is playing against you, trying to increase the number of moves you take by selecting tricky rotations. Does that alter how you think about the game? How you play, in practice?

See, I told you the ceiling could be pretty high.

Make it bigger
Okay, you've solved the 4 coin puzzle, what about 5 coins? Are there interesting versions for larger numbers of coins?

A humble suggestion

Whatever you do, I encourage you to try the 3 coin triangular version with rotations.

Predestination stories (reading lessons)

who: J1 (guest appearances by J2 and J3)
when: early afternoon
where: in the dining room
what material did we use: pack of sight words, Usborne book of Fairy Tales, Jeffrey Archer collection of short stories.

---

We have started to be more consistent about the literacy activities we are doing with the kids.

Warm-ups

We have found very quick warm-up activities worked really well in the math classes we teach at school, so we thought we'd do the same for our literacy sessions. We are still in the process of collecting appropriately short and fun activities, but here is a short list of things we've done so far:

• Sight word sentences: using a pack of sight words, draw 2 and then form a sentence with them. The sillier, the better!
• Sight word story: take 6 or 8 sight words and form a short story with them. 
• Crazy Sentences: reading the strange sentences that come out of this program which was inspired by a game from Peggy Kaye (whose site looks like a good source for other qiuck games).
• Talk about a picture: this is a direct translation of one of our math warm-up activities

Intro to predestination (aka Sleeping Beauty)

For the main event, J1 read (or re-read) the first story in our fairy tale book. As he was reading, I wrote out a couple of questions for us to discuss:

1. What is the location of the story?
2. Who are the main characters (2-4)?
3. What is the conflict in the story?

Of course, these are generic questions we can discuss for almost every story he reads. There were several highlights in our discussion.

What happens with the mean fairy?
In our version, the mean fairy only shows up once, explicitly, in the story, to curse Rose. It is implied in the pictures, that she shows up later to introduce the fated spinning wheel. J1 was strongly drawn to this interpretation based on his own narrative sense of closure and connection. We were a little disappointed that she didn't figure in the ending sequence, but more on that later . . .

Why was Florien successful in rescuing Rose?
Prior to his attempt, several other chaps made an effort and were unsuccessful. J1 said that the reason he was successful seemed to have something to do with Rose drawing Florien's picture earlier in the story and Florien dreaming of Rose. Of course, we don't know how many pictures of princes Rose actually drew, how exact the likeness was to Florien, nor whether the unsuccessful princes had dreamed of Rose or not.

We talked about other stories and came up with these suggestions:

• Maybe Rose and Florien were partners in a prior life (an idea J1 got from the Thai Ramakien)
• Maybe Florien did something nice for the mean fairy and she granted him the ability to rescue Rose. J1's favorite version: The mean fairy turned into a squirrel to run through the forest. She accidentally got caught in a hunter's trap. Florein found her and bandaged her wounds.
• Maybe Florien defeated the mean fairy in battle and won the power to rescue Rose. This version was accompanied by some wild jumping around in a simulated sword (Florien) vs lighting blast (mean fairy) battle.

Predestination vs Free Will
I told J1 about two competing theories: predestination and free will and then we went back to the story to see whether/how each theory was represented. There was a very clear winner, with predestination getting all the points:

• Rose's story told in advance by the curse/blessings of the fairies.
• Rose predicts Florien saving her by her drawing
• Florien predicts saving Rose by his dream
• The other princes fail to save her "just because" (because they weren't fated to do so)
• The king acts to prevent the foretold curse by destroying spinning wheels in the land, but the fate is inescapable

The last point flagged up a classic element of predestination stories: even if the characters take action to change their fate, the results still end up the same. Often, the action taken to prevent the fate is somehow critical in causing it to happen.

Appointment with Death
By chance, I had just read this short story in a Jeffrey Archer anthology. I got it out and we read it together, then talked more about predestination vs free will.

Some other tidbits from the chat:
- who is Death? Why do people anthropomorphize death like this?
- why does the story seem to suddenly shift to first person, form Death's perspective? Is the story more or less clear written this way?

Daily Journal

J1 and J2 both have small notebooks for writing a daily journal. They usually write something about what happened that day, but are free to write whatever they want. Sometimes, it becomes a short story or even a never-ending story.
We talk about what they wrote and then make a vocabulary list related to their note. Usually, it is formed from words they mis-spelled, but the vocabulary words could be things they spelled correctly that highlight interesting patterns or a word they didn't use that is related to the topic.

What about the little one?

For J3, we have been working on phonics and letter recognition. Each evening, we have a focus letter or sound that we ask her to find as we read bedtime stories together. Also, we have been singing phonics songs, particularly ones from this collection: Jolly phonics

Seasonal math and dice, dice, dice

who: J1, J2, J3
what did we use: mangosteen(!), dice, a checkers set.

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Queen of mathematical fruits

My condolences to everyone outside of Thailand, you are current missing some delicious pre-school math: Mangosteen!

The mathematical secrete is on the bottom of the fruit. The flower shape on the bottom has petals that correspond to sections of the fruit inside. This one will have 7 sections

Just to trick you, this is a view from a different one, with 6 fruit sections.

J3 has been having a lot of fun counting the "flower" petals, opening the fruit, then counting the segments to check.

J2 has been using these to practice subitizing (recognizing number groups without counting them). J1 has just been gobbling down the mangosteen.

Dice Games

A dotty dice puzzle
In school for the 2nd and 3rd grades, we introduced a tic-tac-toe variation using dice where the winner is the first player to make a straight line that is a multiple of 5 (more notes here). For homework, the kids played this game with other multiples, in our case 6. During one game with J1, there was an interesting scenario that I saved to discuss with J2. The board has the numbers as marked and the player has just rolled a 4. Where should you play (you can add 4 to any cell of the grid, remember that you don't want your opponent to be able to make a multiple of 6 on their turn):

Race to 20
For J3, we modified one of the games that we had played in the 1st grade class last week. In this game, two players race to be the first to 20 on a 100 board. In this version, both player has a token/marker of their own. For each move, the players roll 2 dice and choose which value to move, then count up their position on the board. First player to 20 or more wins.

The strategy here is very simple, but it was a good game for J3 to start learning about dice and the 100 board. Once she got the point of the strategy, though, she had her own idea:"I'm going to add both dice together and move that amount!"

Snakes and Ladders experiment
J1 and I tried an experiment with snakes and ladders: a race between two different teams.
Team 1: a single marker that has to move according to the roll of one hexahedral die.
Team 2: two markers, the player can choose which one to move on each turn.
The winner is the player who gets a marker to 100.

Which team has an advantage? How big is that advantage? What if you play so that the winning marker has to land exactly on 100 (otherwise bounces back)?

Knights Move

SolveMyMaths recently tweeted about Knights Move (aka Razzle Dazzle.) Since we are always looking out for more/new games to play, it sounded interesting. Also, we could repurpose existing material to play. The game is a cross between chess and basketball. Below, see J1's set-up:

Yes, we are using icosahedral dice as the balls. Also, the chess pieces on the side are spectators, they aren't part of the action (since this is played on a 7x8 board).

One of the most interesting parts was our ability to experiment with rules. Since we didn't have a definitive version, we felt free to try slight modifications. In particular, we experimented with:

• Passing: allowing multiple (chain) passes or just a single pass on a turn.
• Defense: moving into the 8 squares around the ball holder forces them to pass on their turn
• back-and-forth passes: player who passed the ball has to move again before they can receive the ball (we turned the checkers upside down to indicate they couldn't receive a pass). 

Crazy Tangles and the order of the months

who: J1, J2, J3
when; a couple of minutes scattered through the day, 5 minutes at dinner
what did we use: two colored pieces of string (shoelace weight)

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Playing with string

2 years ago, Fawn Nguyen wrote a note about Conway's Rational Tangles. It was a great activity for her algebra class (so go read her note) and I thought our little ones might enjoy it. While getting a soldering iron to fix a toothbrush robot (another upcoming post?) I found two blue and red string that looked really perfect for this investigation.

There are two moves, twist and rotate. Since Fawn does such a fantastic job of showing these moves, I won't even bother trying to describe them. For us, we made it into a two person activity where we hold different strings in each hand. I introduced the ideas in the following sequence:

1. told J1 that there were two moves and showed him. 
2. Then we did a bunch of twists and I counted one each time. 
3. After that, I told him a twist counts as +1. 
4. We did a sequence (3 twists, rotate, one twist, rotate, two twists, rotate, two twists)
5. at that point, he insisted on leading the moves, so he was in charge

J2 had been watching and basically understood the pattern, but we did something similar later as his turn.

J3 had her turn as well and basically just wanted to get tangled up in the string

Reactions
J1 got very excited about making conjectures about what the rotation operation does. His first guess was multiply by -1. We tested with twist, rotate, twist and found that we were back to state 0. However, when we did 2 twists, rotate, then 2 more twists definitely didn't get us to state 0.
Next conjecture was that rotate subtracts 2, but that also is disproven by the fact that TTRTT(0) is not 0. He is still working on it.

For J2, he liked to get complicated and started out with 10 twists and then a rotate. From that point, he let me call the moves (TRTTRTTRTTRTTRTTRTTRTTRTTRTT) and was totally delighted to see us get untangled back at state 0 after all of that.

J3 got a lot of mileage out of disrupting other people's tangles. This was certainly fun for her, but a bit frustrating for the older two.

Which month is first?

At dinner, I asked the kids which is first, January or February? J1 and J2 quickly said "January."
Next question, which is first, January or March? January again.
Next, January or April? Again, quick answer of January. P interjects, is that going to last?
January or May? J2 again says "January" quickly, J1 is starting to hesitate.
January or July? January again from J2, July from J1.

Why?
J2 is pretty clear; "January is the first month of the year, so it always comes first."
J1 has a different idea. It is June now, so our next July will come before the next January. Thus, July is first.

Next question, which comes first, January or December?
J2 again says "January, it is the first month of the year, so it has to come first."

What month comes immediately before January?
J2 says "December."

Note :J3 doesn't really know the months yet, so she wasn't very proactive, but she did enjoy repeating the others' answers.

This question occurred to me when reading Who's the Oldest over at Musings of a Mathematical Mom.