## Thursday, February 23, 2017

### A bit of 3D(ish) geometry

Note: this post started out focused on two recent geometry projects. However, the Desmos Function Carnival, which I originally just included as a miscellaneous item, is also worth your time.

# Nets and solids

The most ambitious project recently was led by P (the mom, of course). She found nets for 3-d shapes and supervised J1 and J2 as they created a nice display to take to school.

Here are the nets: SenTeacher Polyhedral nets. They have a collection of other printables, but this collection of nets seems to be the most interesting. Take a look and let me know if you see anything else worthwhile.

Here's the completed board:

A profile picture to show that these are really 3-d:

Solid Shapes and Their Nets (I think Jo Edkins is the author) has a nice discussion of nets and a little puzzle game to distinguish nets that fold into the platonic solids and which don't. Feel free to try to do this for the icosahedron or dodecahedron!

# A two dimensional challenge?

J2 asked me about a triangle with two right angles. Of course, we all know that a triangle can't have two 90 degree angles, right? Well, this fits in my list of math lies from this old post: 23 isn't prime.

We talked briefly about triangles on a plane and agreed that two 90 degree angles wouldn't work. If we try by starting with a side and building two right angles on that side, we just get parallel lines. Ok, that's standard.

We discussed spherical geometry for a bit. Not shown is our first attempt on the other side of the tennis ball that did have two right angles, but the third wasn't. He wanted to see an equiangular triangle on a sphere. This led to further discussion of life on a sphere:

• what is the largest interior angle sum for a triangle?
• what is the smallest interior angle sum?
• are there any parallel lines?
• are there any squares? are there even any rectangles?
• is π (the ratio of the circumference to diameter of a circle) a constant?
• do we have to think like this in real life because the earth is close to a sphere?
Picking up the point about π, it occurs to me that this is another math lie. This point got a nice treatment recently by SMBC:

 I made this pic small so you will go to the site and look at Zach Weinersmith's other awesome work
Some open follow-ups:
• what is an equiangular quadrilateral on the sphere?
• what about hyperbolic geometry? I think no triangles with two right angles. If I recall correctly, triangles there have angle sums smaller than 180 degrees (or π radians, ha!)

# Other Misc Math

Now for my usual grab bag of other things we've been doing. Some of these are really great activities, so don't skimp on this section!

Desmos Function Carnival
We learned about the Desmos Function Carnival activity from a post by Kent Haines which, in turn, we learned about via Michael Pershan's post. They both were writing about teaching functions to their students and have outlined a really nice sequencing of lessons, if you're into that kind of thing.

In case you are, you might like to know that there is another flavor of Function Carnival available through the Desmos teacher site with 2 other activities. Each version is worth checking out because both of worthwhile activities that aren't in the other. Also, Kent links to this nice graphing activity from the Shell Center which also worked well with J1 and J2.

For our purposes, I was more interested in the mathematical physics (plots of position vs time or velocity vs time) and the fun of creating wacky animations with impossible plots.

If I can figure out how to post the animations, I will update this page, since the animations really enhance the experience. For now, let me give you some screen caps of some of their proposed graphs from the Cannon Man (height) activity:

 Have a free-form drawing tool? Make your name!

 Note: the vertical stripe doesn't really work with this animation, but the patriotic spirit is there!

 Our first attempt at Quantum Field Theory

Here is J2 playing with the function carnival, but J3 (4 years old) enjoyed it just as much:

Different representations
J3 is working on place value and playing with different representations of numbers. Here, she's got the 100 board, an abacus, and foam decimal models. Sometimes I challenge her to make a number, sometimes she challenges me.

## Tuesday, February 21, 2017

### RSM International Math Contest

Sometime the first week of February, the RSM offers an on-line math competition. For the second year, I had J1 and J2 work through the problems at grade 3 and grade 4 level.  This post is about our thoughts on competition, but I end with a problem from the contest that had us debating.

# Compete and win!

To get it out of the way, let me explain my interest in the competition and why I had the kids enter. Above all, I was curious about the problems and expected they would be interesting challenges. We found the puzzles last year interesting, so I was pretty sure this year's collection would also be nice.
While I know that there are tons of excellent activities, I couldn't resist making use of this resource when someone thoughtful had put together a convenient collection in one place.

Second, I am curious about levels and assessments. What types of questions does RSM think 3rd and 4th graders should be able to answer, but find challenging? How difficult would our two little ones find them? Since we are currently operating outside a standard curriculum framework, it is hard for me to judge where they are or what we should expect of them. Admittedly, there are other ways I could form this assessment, but they take more effort.

Notice that I don't really care about how well they perform on the test. It is interesting information for me, but I don't need them to do well.

P has a different view. She was an olympiad kid and sees competitions as the easiest way for our kids to distinguish themselves. You know the anxiety: if they don't win competitions, they won't get into Harvard, they'll end up on the street somewhere.

Perhaps unsurprisingly, the kids are getting a mixed message about the importance and reason for participating in competitions. This year, J1 was particularly sensitive. He was very resistant to doing the RSM test. I spent a lot of time talking with him. Mainly, I wasn't hoping to convince him to do the test, but I wanted to explore other issues related to the expectations he feels on himself, his relationship with J2, and his mindset about his own learning.

I am trying to communicate:

1. effort and progress are important, starting point and base ability aren't. I know this is debatable, but I'm talking about the differences between my kids where I'm on pretty firm ground.
2. there are many aspects to math and mathematical ability. Calculating and answering questions quickly are facets, logical reasoning, strategic thinking, spatial reasoning, asking good questions, gathering data, exploring connections, etc, etc are all components, too.
3. time isn't important. One issue with the RSM test is that it is timed. This goes in the face of our repeated efforts to emphasize that the time it takes to solve something is not a key consideration. To help with this, I screen capped the questions so he could work on them after the official time had expired. Note, however, his recorded performance on the test was based on work within the contest rules.
The next section might help illustrate that last point.

# Which area do we want?

As mentioned above, one of the questions from grade 4 cause us to argue among ourselves. I'm paraphrasing slightly:
Two tennis ball machines stand on opposite ends of a 25 meter by 10 meter court. The yellow machine shoots yellow balls that stop on the court 2 meters to 16 meters from the yellow machine's side. The green machine shoots green balls that stop on the court 5 meters to 20 meters from the green machine's side. Find the area of the court that has balls of either color on it.
Here are some ideas from our discussion:

• Option A: we want to find the area of overlap, because that's the only place we could find balls of either color.
• Option B: we want to find the combined areas, because that's where we could find tennis balls, either color.
• Option 1: the area with green balls is a rectangle, the area with yellow balls is a rectangle.
• Option 2: think of the ball machines as single points, shooting balls at various angles and various distances. the target area for each machine is an intersection of of a circular ring and a rectangle. This idea came from J2 when we were looking over the grade 4 questions after time had expired.
To answer the question, choose either A xor B and choose either 1 xor 2. So, what is the right answer?

I think the right answer is to have this discussion, to encourage multiple interpretations (with justification), to see how the answers compare, to think about what other things we could do to make the intended interpretation more clear (a diagram seems the most obvious), to recognize that this is part of the richness of math and it can't be represented by a single numerical answer on a timed test.

### Cryptarithmetic Puzzles for Grades 1 to 4

Inspired by a series of puzzles from Manan Shah, I decided to have the kids play with cryptarithmetic puzzles today. In addition to borrowing some of Manan's puzzles, I also used some from this puzzle page: Brain Fun. I've included some more comments below about the Brain Fun puzzles.

My main concern was whether the puzzles were at the right level. In particular, I was afraid that the puzzles would be too hard. In fact, I tried solving a bunch of them yesterday and actually found myself struggling. I'll ascribe some of that to being tired and sick. However, my intuition was to make some simpler puzzles of my own. In particular, I added:
• puzzles that have many solutions: I figured that many solutions would make it easy to find at least one.
• a puzzle that "obviously" has no solution. Now, obviously, the word "obviously" is a sneaky one in math, but I was pretty sure the kids could see the problem with this structure.

For the younger kids, I started with a shape substitution puzzle. This is one our family explored almost 2 years ago: Shape Substitution. I don't recall the original source.

Two reasons why I started with this. First, it has a lot of solutions, but there is an important insight that unlocks those solutions. Second, by using shapes, we can write possible number solutions inside them as we solve or guess-and-check the puzzle. This made it easier for the kids to see the connection that all squares have the same value, etc.

The second puzzle: BIG + PIG = YUM
Really just a warm-up practicing the rules and doing a little bit of checking that we haven't duplicated any numbers.

The third puzzle: CAT + HAT = BAD
Again, lots of solutions, but noticing leads to a good insight.

This is a trick puzzle. The kids know that I like to tease them, so they are aware they need to look out for things like this. We discussed this in class and I suggested they give this puzzle to their parents.

Fifth puzzle: CURRY + RICE = LUNCH
When I translated this to Thai, all the kids laughed. I was sneaking a little bit of English practice into the lesson and then they realized that it was worth trying to read all the puzzles, not just solve them.

Sources: I think I made up all of these puzzles (original authors, please correct me if I'm wrong).

The older kids already had experience with these puzzles. We did refresh their memory a bit with BIG + PIG = YUM

I asked them to give me the rules and explain why those rules made sense. As with most games, I want to communicate that we're doing things for a reason, but those reasons can be challenged. If they think it makes sense to do it a particular way, we're open to their ideas.

Same discussion as for the younger kids. When prompted, this was pretty easy for them to spot, but they weren't naturally attuned to think about whether a puzzle had solutions or how many. This led me to take a vote on all the puzzles at the end to see who thought the puzzles would have 0, 1 or many solutions.

Third puzzle: ALAS + LASS + NO + MORE = CASH
A puzzle from Brain Fun. I think this is one of the easier ones on that page. Again, a bit of English practice.

Fourth puzzle: LOL + LOL + LOL + .... + LOL = ROFL (71 LOLs)
This was from Manan. I think it is one of the easier ones in his collection, but it looks daunting. Turns out none of the kids in the class were familiar with (English) texting short-hand, so my attempt to be cool fell flat.

Fifth puzzle: CURRY + RICE = LUNCH
Again, everyone was delighted when I translated this one. We're in Thailand, after all, so at least one puzzle had to be about food.

# The key exercise

The final assignment everyone (all four grades) was given was to make up a puzzle for me to solve. I was thinking it would be nice to have one in Thai, but we decided to keep it in English as further language practice.

Manan wrote a nice post about having kids design their own puzzles. If it goes well, this is actually the activity that ties a lot of the learning messages together: they think about structure, they think about what allows multiple or single solutions, they apply their own aesthetic judgment, they use their knowledge of the operations, they are empowered with an open-ended task that cannot be "wrong."

We'll see how it goes. At the very least, I expect a lot of work for myself when their puzzles come in!

An extra sweetener
Two kids asked if we could use other operations than addition. That prompted me to put this on the table (also from Brain Fun):

DOS x DOS = CUATRO

# Brain Fun Problems

The first time I'd seen the Brain Fun problems, I added them to a list and called them "basic" (see this page.) When I actually went to solve them, though, they didn't seem so easy.

Big confession time: I actually looked at some of the solutions.  However, I was disturbed to see that the solutions involved extra information that wasn't included as part of the problem statement! For example, in THREE + THREE + FIVE = ELEVEN, the solution assumes that ELEVEN is divisible by 11. This seems to be the case for several of the puzzles involving written out arithmetic:

TWO + TWENTY = TWELVE + TEN (assume 20 divides TWENTY and 12 divides TWELVE, I wasn't clear about whether any divisibility was assumed for TWO and TEN)

I'm not sure if similar assumptions are allowed/required for any of the others.

Maybe I shouldn't complain, since this assumption creates an additional constraint without which there could be further solutions. Perhaps part of the reason it doesn't sit well is aesthetic. In the 3 + 3 + 5 = 11 puzzle, 3 doesn't divide THREE and 5 doesn't divide FIVE.

Lastly, there is a typo in the final puzzle of the Brain Fun page. That puzzle should be
TEN x TEN = FIFTY + FIFTY

## Thursday, February 16, 2017

### More Man Who Counted (gaps and notes)

As previously mentioned, we have been reading The Man Who Counted. While the story is good and there are nice math puzzles, we've found some of our best conversations have come from errors or weaknesses in the book. Here are three examples:

# How old was Diophantus?

In chapter 24, we encounter a puzzle to figure out how old Diophantus was when he died. In summary, the clues are:

1. he was a child for 1/6 of his life
2. he was an adolescent for 1/12 of his life. (J1: "what's that?" J0: "a teenager")
3. childless marriage for 1/7 of his life
4. Five more years passed, then had a child
5. The child got to half its father's age, then died.
6. Diophantus lived for four more years
Perhaps we are wrong about our interpretation of the clues, but we noticed two things:
(a) the answer is not a whole number of years.
(b) the answer given in the book doesn't fit the clues.

For the first part, it seems a natural assumption of these types of puzzles that we are only working with whole number years. Sometimes, this is an interesting assumption to directly challenge.
Here, since the clues involve a second person (Diophantus's child) we felt whole numbers were a strong assumption. Also, the name Diophantus, you know?

Each clue required some discussion for us to agree on the interpretation. The one that seems most open is the fifth clue. In particular, did the child live until its age was half of the age of its father at the time of birth or to the point that, contemporaneously, it was half its father's age?

For completeness, I'd note that neither interpretation matches the book's answer. The first interpretation does allow a whole number answer, but it doesn't give whole numbers for all the listed segments of Diophantus's life.

Just so you can check for yourself, the solution given in the book is 84 years old.

How do you fix it?
We discussed several possible fixes:

• accept answers that aren't whole numbers or require whole number segments for each clue. This allows us to take the alternative interpretation of the fifth clue (though that still isn't satisfying) or to accept the clues and just take a new answer. This isn't satisfactory because... Diophantus.
• Change clue 4 or clue 5 to match the book's answer. This approach seemed to fix the puzzle without distorting it or changing the mathematics required to analyze it.
• Change clue 1, 2, or 3. While possible, these seemed to open the possibility of changing the character of the puzzle. Also, these fractions were plausible based on our own experience of human life spans.
Of course, an even more satisfying answer would be to introduce a further variable and make the puzzle into one that makes heavy(ier) use of the integer restriction.

# Clever Suitors

In chapter 31, Beremiz is confronted by a nice logic puzzle. Three suitors are put to a test, each is blindfolded and has disc strapped to his back. The background of the discs: other than color, the discs are all identical, there are five to choose from, 2 black and 3 white.

The first suitor is allowed to see the colors of the discs on the backs of his two competitors, then required to identify the color of his own disc and explain his reasoning. He fails and is dismissed.

The second suitor is allowed to see the disc on the back of the third suitor, then required to identify the color of his own disc and explain his reasoning. He fails and is dismissed.

Finally, the third suitor is required to identify the color of his own disc and explain his reasoning.  He succeeds.

Weakness 1
As a logic puzzle, we enjoyed this. Our problems came from the context in the story. This challenge was set to the three suitors as a way of fairly judging between them by finding the most clever suitor. However, this process was clearly unfair. In fact, it is inherent in the solution that it was impossible for the first and second suitors to determine the color of their own discs.

This led to a nice discussion about who really held the power in this process: the person who structured the problem by deciding what color disc should be on which suitor and what order they would be allowed to give their answers.

Extensions:
• consider all arrangements of discs. Are there any arrangements where none of the suitors can answer correctly?
• What is the winning fraction for each suitor? If you were a suitor, would you prefer to answer first, second, or third?

Weakness 2
Our second objection was non-mathematical, but again related to the story context. The fundamental problem wasn't how to choose a suitor. The fundamental problem was how the king could remain peacefully friendly toward all the suitors' home nations through this process.

For this discussion, we went back to the story of Helen of Sparta, which we'd read a long time ago in the D'Aulaire's Book of Greek Myths. Of course, that also led to discussion of the division of the golden apples, another puzzle we all felt surely could have been solved more effectively with some mathematical reasoning...

# The Last Matter of Love

The last puzzle of the book is in chapter 33. It is another logic puzzle, again intended to test the merit of a suitor in marriage. The test:

• there are five people
• two have black eyes and always tell the truth
• three have blue eyes and always lie
• the suitor is permitted to ask three of them, in turn, a "simple" question each.
• the suitor must determine the eye color of all five people
As a logic puzzle, we readers get some extra information:
1. The first person is asked: "what are the color of your eyes?" The answer is unintelligible.
2. The second is asked: "What did the first person say?" The answer is "blue eyes."
3. The third is asked: "What are the eye colors of the first and second people?" The answers are "the first has black eyes and the second has blue eyes."
Simple questions
Our first objection was the part about asking "simple" questions. Having developed our taste for these types of puzzles through the knights and knaves examples of Raymond Smullyan (RIP, we loved your work!!!), the third question really bothered us. If you're going to go that far, why not ask the third person for the color eyes of all five people?

Personally, I would prefer that the puzzle require us to ask each person a single yes/no question.

As an extension: can you solve the puzzle with that restriction?

Getting lucky
Again, we felt that this puzzle didn't meet the requirements of the context: to prove the worthiness of the suitor. Putting aside the question of whether this is really an appropriate way to decide whether two people should be allowed to marry, the hero here got lucky.

Extension: what eye color for the third person would have caused the suitor to fail?
Extension: what answer from the third person would have caused the suitor to fail?
Extension: for what arrangement of eye colors would the questions asked by the suitor guarantee success?
Extension: what was the suitors' probability of success, given those were the three questions asked?

Waste
Our final objection was the simple waste in the first question. From a narrative perspective, this is justified and even seems made to serve the purposes of the suitor. However, it opens another idea:
can you solve the puzzle, regardless of eye color arrangement, with only two questions?

Feel free to test this with yes/no questions only or your own suitable definition of a "simple" question.

# The power of...

As a final thought, let me say that I think errors and ambiguity in a text are a feature, not a bug. It is another great opportunity for us to emphasize that mathematics is about the power of reasoning, not the power of authority.

## Tuesday, February 14, 2017

Recently, we have been playing the following games:

1. Go (baduk, weiqi, หมากล้อม). For now, we are playing on small boards, usually 5x5 or smaller.
2. Hanabi
3. Cribbage
4. Qwirkle (not regulation play, a form of War invented by J3 and grandma)
5. Munchkin
6. UNO
7. Vanguard
I've ordered these by my own preference. In fact, I would be delighted playing just the first two exclusively and am happy to play cribbage or Qwirkle when asked.

For the other three, I find myself biting my tongue a bit and grudgingly agreeing to be part of the game. I'm in the mood for strategic depth and a moderate (but not large) amount of pure chance. Part of my feeling was echoed in a recent My Little Poppies post: Gateway Games.

However, as in the MLP post, I recognize that my enjoyment of the game is only a part of the reason for the activity. I guess the kids' enjoyment counts, too.

Beyond that, even the games with limited depth are helping to build habits and skills:
• executive control: assessing the situation, understanding what behavior is appropriate, understanding options and making choices.
• general gaming etiquette: taking turns, use of the game materials
• meta-gaming: helping and encouraging each other, making sure that the littler ones have fun, too
• numeracy and literacy: every time a number or calculation comes up or when something needs to be read, they are reinforcing their observation that math and reading are all around them.
• meta-meta gaming: game choice, consensus building, finding options that interest and are suitable for all the players, knowing when it is time to play and when it isn't.
As a family, and a little team, they are also building a shared set of experiences and jargon as they absorb ideas from each of the games.

All of these are, of course, enough reason to make the effort to be open minded and follow their gaming lead.

## Monday, February 13, 2017

### NRICH 5 Steps to 50

A quick note about the game we played in first grade today: 5 Steps to 50.

This is an NRICH activity that I've had on my radar for a while. I even made a pencilcode program to explore the activity in reverse. True to their other activities (check them out!!!) 5 steps to 50 requires very little explanation, is accessible to students with limited background, but has depth and richness.

Our lesson outline
I explained the basic activity and did an example at the board. To get my starting value, I had one student roll for the 10s digit and one for the 1s digit. Then we talked through together as we added 10s and 1s.

I then distributed dice and had the kids try 3 rounds. As they worked, I confirmed several rules:

1. the only operations allowed are +1, -1, +10, -10
2. we must use exactly five steps (I note that this is ambiguous on the NRICH description, they say "you can then make 5 jumps")
3. we are allowed to do the operations in any order
4. we can mix addition and subtraction operations
After everyone had been through 3 rounds, we regrouped to summarize our findings:

We helped the kids resolve disagreements and then posed the following:
For those to challenges, we kept the restriction that the numbers must be possible to generate from 2d6.

Basic level
To engage with the activity, some of the kids just started trying operations without much planning. This quickly reinforced the basic points about addition and place-value and commutativity of addition.

For these kids, it was helpful to ask a couple of prompting questions:

• What do you notice? This is a standard that never gets old!
• If this path doesn't get to 50, does that mean there is no path to 50?
This second question, particularly, raises the interesting observation that it is easy to show when a number can jump to 50 (just show a path) but to show that no path is possible requires a different type of thinking.

The next level of sophistication was really about noticing that the key consideration is the distance to 50. In particular, this identified a symmetry, where n could jump to 50 if 2*50 - n can jump to 50. Of course, the kids didn't phrase this relationship in this way....

The next major step is thinking about a way to systematically write down the paths.

## Wednesday, February 1, 2017

### Perfect Play for My closest neighbor

Joe Schwartz at Exit10a wrote a fraction comparison post that prompted me to write up more of my experience and thoughts on this game.

Let's find perfect play
This week, I intended to use the game one last time with the 4th graders as an extended warm-up to our class. The challenge I presented:

If we got super lucky and were given perfect cards for each round of the game, what are the best possible plays?

My intention was to spend about 20 minutes on this. Depending on how quickly it went and the kids' reactions, I considered giving them a follow-up for a short homework: what are the best plays if we include all cards A (1) through K (13)?

How did it go?
In the end, the basic activity took the whole class. These comparisons were difficult for the kids, so we spent time talking about each different strategy for comparison:

1. common denominators
2. common numerators
3. distance to 1
4. relationship to another benchmark number. Like 1/2 in Joe's 4/6 and 8/18 example, a benchmark is a "familiar friend" that should be relatively easy to see it is larger than one and smaller than another. In practice, 1/2 seems to be the most popular benchmark.

For visualization, drawing on a number line seemed to work best.

I did not assign the full deck challenge as homework. Instead, we gave them some more work with fractions of pies and bars.

What have I learned?
This game is really effective at distinguishing levels of understanding:
(0) some kids are totally at sea. They don't really understand what this a/b thing means, how a and b are related, etc. These kids struggle with the first round of the game when the target is 0, when the idea is to just want to make their fraction as small as possible.

(1) Some kids have got a basic understanding of the meaning of the fraction and can play confidently when the target is 0 or 1. They might still be weak about equivalent fractions. Trying to play some spot-on equivalents when 1/3 and 1/2 are targets is a give-away.

(2) familiar with some frequent friends: kids who can tell readily whether their plays are larger or smaller than the target for 1/3, 1/2, 3/4.

(3) proficient: have at least one consistent strategy they can work through to make a comparison

(4) fraction black-belts: using multiple strategies, already familiar with many of the most common comparisons.

What would I do differently?
Generally, I think it is valuable to spend more time and more models directed at the basic understanding of what fractions mean. The kids who were at or close to stage 4 have, over the years, been seeing diagrams of pies, cakes, chocolate bars, number lines and physical experience with baking measures and fractional inches on measuring tapes and rulers. Oh, and also actual pies (mostly pizza), cakes, cookies, and chocolate bars discussed using fractional language.

More locally, for this game in a class of mixed levels, I would

• lean toward doing this more as a cooperative puzzle
• re-order the targets for the rounds as 0, 1, 1/2, 3/4, 1/3, 2 (note: I don't have strong feelings about where 2 fits in this sequence)
• I also would consider allowing equivalent fractions to the target as winning plays

### Impassable Din Daeng (BKK intersections 3)

It has been a while since I've done one of these.  For your topological and civil engineering pleasure, I present  Din Daeng intersection (แยกดินแดง):

 Notice how the expressway obscures key details?

 Magnified view. We now see a tunnel, but where does it emerge?

This intersection has a special place in my heart. Last month, J1 and J2 played in a squash tournament at the Thai-Japan Youth Centre. It isn't visible on either map I've included, but is just to the northeast of the intersection.  Since we live on the west of the intersection, we needed to cross somehow.

After three failed attempts (following the driving directions on google maps) to get through from west to east, I ended up parking our car in one of the small side streets and we just walked. The walk took about 30 minutes...

### ideas for upcoming classes

#### warm-ups for all

WODB: (1) shapes book (2) wodb.ca
any: Traffic lights/inverse tic tac toe/faces game
good options here, mostly grades 1/2: some games
dots & boxes (maybe with an arithmetic component)
loop-de-loops

close to 100 game:
Equipment: A pack of cards with 10 and face cards (J,Q,K) removed.
Procedure:
- Deal out 6 cards to each player
- Each player picks 4 cards from the 6 cards they were dealt to form a pair of 2-digit numbers.  The goal is to get the sum of the two numbers as close to 100 as possible but cannot exceed 100.

Factor finding game (maybe warm-up?)
Factors and Multiples game
Contig for 3 and 4 (explanation).
Times tic-tac-toe: review for Grades 3 and 4
Fraction war for grade 4 (smallest card is numerator)
Multiplicaton models: worth making for grades 3-4 for solidifying concepts? Associated games
d 2

Pico Fermi Bagel

Magic triangle puzzles

damult dice

(1) Dice game perudo
Equipment
- multi-player, 2-5
- Everyone gets the same number of 6 sided dice (full game they get 5, I would start with 3)
- Everyone has a cup to shake and conceal their dice

Basic Play
- Simultaneously, players shake their cups and turn them over on the ground or a table. They peak in to look at their own dice, but keep them concealed from the other players.
- starting randomly (or from the person who lost a dice in the last round), players make bids, for example: two 3s.
This bid signifies that the player has 2 (or more dice) showing the value 3.
- the next player has two choices:
• call/doubt the previous player's bid: if they do this, all players show their dice. If there are enough to meet the bid, the caller loses a die. If not, then the bidder loses a die.
• raise, either the number of dice or the value or both get increased
- Ones are wild, they count as any number toward the target bid
- If someone drops to their last dice, they start the next round. On that round, only the number of dice can be increased in the bid, not the value. Ones are not wild on this round
- After someone bids, the next player has a third option, to call "exact." If the bid is exactly matched by the dice, then the bidder loses a die and the caller gets an extra one. If the actual dice show either more or less than the bid, the caller loses a die.

remainder jump
we played this game before, but we could give them blank boards and let them create. See the last page here: http://ba-cdn.beastacademy.com/store/products/3C/printables/RemainderJump.pdf

another good game:

We should still use the pyramid puzzle sometime:

(1) double digit and double dollar:
We've done something like this, but I think there could be a good variation done trying to make 1000 baht, using 1, 2, 5, 10, 20, 50, and 100 baht units.

(2) biggest rectangle. This could be used as a warm-up. For the older kids, they will probably have seen something like this, but I like the inclusion of perimeters that are even but not divisible by 4 and odd perimeters and the question about "smallest area" (here are 5 questions).

(3) some of these games are promising: